Recent zbMATH articles in MSC 11https://zbmath.org/atom/cc/112022-09-13T20:28:31.338867ZUnknown authorWerkzeugApproximation methods in science and engineeringhttps://zbmath.org/1491.000032022-09-13T20:28:31.338867Z"Jazar, Reza N."https://zbmath.org/authors/?q=ai:jazar.reza-nPublisher's description: \textit{Approximation Methods in Engineering and Science} covers fundamental and advanced topics in three areas: Dimensional Analysis, Continued Fractions, and Stability Analysis of the Mathieu Differential Equation. Throughout the book, a strong emphasis is given to concepts and methods used in everyday calculations. Dimensional analysis is a crucial need for every engineer and scientist to be able to do experiments on scaled models and use the results in real world applications. Knowing that most nonlinear equations have no analytic solution, the power series solution is assumed to be the first approach to derive an approximate solution. However, this book will show the advantages of continued fractions and provides a systematic method to develop better approximate solutions in continued fractions. It also shows the importance of determining stability chart of the Mathieu equation and reviews and compares several approximate methods for that. The book provides the energy-rate method to study the stability of parametric differential equations that generates much better approximate solutions.
\begin{itemize}
\item
Covers practical model-prototype analysis and nondimensionalization of differential equations;
\item
Coverage includes approximate methods of responses of nonlinear differential equations;
\item
Discusses how to apply approximation methods to analysis, design, optimization, and control problems;
\item
Discusses how to implement approximation methods to new aspects of engineering and physics including nonlinear vibration and vehicle dynamics.
\end{itemize}Book review of: B. Poonen, Rational points on varietieshttps://zbmath.org/1491.000082022-09-13T20:28:31.338867Z"Baxa, C."https://zbmath.org/authors/?q=ai:baxa.christophReview of [Zbl 1387.14004].Book review of: A. Stakhov, Mathematics of harmony as a new interdisciplinary direction and ``golden'' paradigm of modern science. Volume 1: The golden section, Fibonacci numbers, Pascal triangle, and Platonic solidshttps://zbmath.org/1491.000212022-09-13T20:28:31.338867Z"Rindler, H."https://zbmath.org/authors/?q=ai:rindler.haraldReview of [Zbl 1444.11001].Productive anachronism: on mathematical reconstruction as a historiographical methodhttps://zbmath.org/1491.010582022-09-13T20:28:31.338867Z"Schneider, Martina R."https://zbmath.org/authors/?q=ai:schneider.martina-rAuthor's abstract: The spectrum of practises of mathematical reconstruction is explored on the basis of a case study on a partly successful mathematical reconstruction of the Chinese Remainder Theorem. In the 19th century L. Matthiessen reconstructed two versions of this theorem on the basis of a corrupted secondary source concerning ancient Chinese mathematics. He identified the more restricted version of the theorem with a Gaussian approach, whereas the other more general one was described as something new surpassing contemporary mathematical European achievements. I identify and compare two different types of mathematical reconstructions in Matthiessen's contributions, and explore their historiographic functions. To capture the relation between mathematical reconstruction and anachronism, the time scheme in the case study is analyzed and linked to the concept of pluritemporality according to Landwehr. This more complex perspective on the category of time in historical research suggests that anachronism should be re-conceptualized. It allows for a discussion of some of the conditions under which mathematical reconstructions can be used in a historiographically sensitive way in a different setting. I argue that this kind of historiographically sensitive mathematical reconstruction can be regarded as a productive historiographical method.
Reviewer's remarks: The author's text represents Chapter 5 of the book [\textit{N. Guicciardini} (ed.), Anachronisms in the history of mathematics. Essays on the historical interpretation of mathematical texts. Cambridge: Cambridge University Press (2021; Zbl 1471.01005)]. It is the (or perhaps better: a) story about what people in the last two centuries wrote around the history, and the description of the ``dayan rule'', i.e. the Chinese Remainder Theorem in number theory.
The author provides in extenso what the title of Chapter 5 has to mean, by analysing such things as ``who was first with what?'' and how methods thereoff influenced mathematics understanding, over and over, and on different moments over the ages.
The work of L. Matthiessen is taken as a paradigma example of the ideas of historiography in mathematics more in particular his writings on the Chinese Remainder Theorem.
The contents of Chapter 5 are worthwhile to read on his own account. On the other hand, being a chapter of a recent published book, the interested connaisseur is almost obliged to buy the book. As to this referant, I would say: why not?
It is worth to mention that Chapter 5 does contain a corresponding list of references; in it one finds contributions in relevance, for instance
[\textit{K. L. Biernatzki}, J. Reine Angew. Math. 52, 59--94 (1856; ERAM 052.1367cj);
\textit{V. Blåsjö} and \textit{J. Hogendijk}, ISIS 109, No. 4, 774--781 (2018; Zbl 1411.01034);
\textit{J. Bell} and \textit{V. Blåsjö}, Math. Mag. 91, No. 5, 341--347 (2018; Zbl 1407.40001);
\textit{M. Cantor} (1894); \textit{P. E. Dickson} (1921);
\textit{M. N. Fried} (2018);
\textit{J. P. Hogendijk} (1989);
\textit{A. Landwehr} (2013);
\textit{U. Libbrecht}, M.I.T. East Asian Science Series. Vol. 1. Cambridge, Mass.-London: The M.I.T. Press (1973; Zbl 0291.01003);
\textit{Y. Mikanic} (1913); \textit{T. J. Stieltjes} (1890); \textit{A. Wylie} (several years in the 19th century)],
etc.
For the entire collection see [Zbl 1471.01005].
Reviewer: Robert W. van der Waall (Huizen)The elements of advanced mathematicshttps://zbmath.org/1491.030022022-09-13T20:28:31.338867Z"Krantz, Steven G."https://zbmath.org/authors/?q=ai:krantz.steven-georgePublisher's description: This book has enjoyed considerable use and appreciation during its first four editions. With hundreds of students having learned out of early editions, the author continues to find ways to modernize and maintain a unique presentation.
What sets the book apart is the excellent writing style, exposition, and unique and thorough sets of exercises. This edition offers a more instructive preface to assist instructors on developing the course they prefer. The prerequisites are more explicit and provide a roadmap for the course. Sample syllabi are included.
As would be expected in a fifth edition, the overall content and structure of the book are sound.
This new edition offers a more organized treatment of axiomatics. Throughout the book, there is a more careful and detailed treatment of the axioms of set theory. The rules of inference are more carefully elucidated.
Additional new features include:
\begin{itemize}
\item An emphasis on the art of proof.
\item Enhanced number theory chapter presents some easily accessible but still-unsolved problems. These include the Goldbach conjecture, the twin prime conjecture, and so forth.
\item The discussion of equivalence relations is revised to present reflexivity, symmetry, and transitivity before we define equivalence relations.
\item The discussion of the RSA cryptosystem in Chapter 8 is expanded.
\item The author introduces groups much earlier. Coverage of group theory, formerly in Chapter 11, has been moved up; this is an incisive example of an axiomatic theory.
\end{itemize}
Recognizing new ideas, the author has enhanced the overall presentation to create a fifth edition of this classic and widely-used textbook.
See the reviews of the first, second and third editions in [Zbl 0860.03001; Zbl 0988.03002; Zbl 1243.03001]. For the fourth edition see [Zbl 1375.03002].A proof of Bel'tyukov-Lipshitz theorem by quasi-quantifier elimination. I: Definitions and GCD-lemmahttps://zbmath.org/1491.030112022-09-13T20:28:31.338867Z"Starchak, M. R."https://zbmath.org/authors/?q=ai:starchak.m-rSummary: This paper is the first part of a new proof of decidability of the existential theory of the structure \(\langle \mathbb{Z} ; 0, 1, +, -, \leq , \vert \rangle \), where \(\vert\) corresponds to the binary divisibility relation. The decidability was proved independently in 1976 by \textit{A. P. Bel'tyukov} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 60, 15--28 (1976; Zbl 0345.02035)] and \textit{L. Lipshitz} [Trans. Am. Math. Soc. 235, 271--283 (1978; Zbl 0374.02025)]. In 1977, \textit{V. I. Mart'yanov} [Algebra Logic 16, 395--405 (1977; Zbl 0394.03038); translation from Algebra Logika 16, 588--602 (1977)] proved an equivalent result by considering the ternary GCD predicate instead of divisibility (the predicates are interchangeable with respect to existential definability).
Generalizing in some sense the notion of quantifier elimination (QE) algorithm, we construct a quasi-QE algorithm to prove decidability of the positive existential theory of the structure \(\langle \mathbb{Z}_{ > 0} ; 1, \{ a \cdot \}_{a \in \mathbb{Z}_{ > 0}}, \operatorname{GCD}\rangle \). We reduce to the decision problem for this theory the decision problem for the existential theory of the structure \(\langle \mathbb{Z} ; 0, 1, +, -, \leq , \operatorname{GCD}\rangle \). A quasi-QE algorithm, which performs this reduction, will be constructed in the second part of the proof.
Transformations of formulas are based on a generalization of the Chinese remainder theorem to systems of the form \(\operatorname{GCD} (a_i, b_i + x) = d_i\) for every \(i \in [1..m]\), where \(a_i\), \(b_i\), \(d_i\) are some integers such that \(a_i \ne 0\), \(d_i > 0\).A proof of Bel'tyukov-Lipshitz theorem by quasi-quantifier elimination. II: The main reductionhttps://zbmath.org/1491.030122022-09-13T20:28:31.338867Z"Starchak, M. R."https://zbmath.org/authors/?q=ai:starchak.m-rSummary: This paper is the second part of a new proof of the Bel'tyukov-Lipshitz theorem, which states that the existential theory of the structure \(\left\langle \mathbb{Z};0,1, + , - , \leqslant ,| \right\rangle\) is decidable. We construct a quasi-quantifier elimination algorithm (the notion was introduced in the first part of the proof) to reduce the decision problem for the existential theory of \(\left\langle \mathbb{Z};0,1, + , - , \leqslant , \operatorname{GCD} \right\rangle\) to the decision problem for the positive existential theory of the structure \(\left\langle \mathbb{Z}_{ > 0};1,\{ a \cdot \}_{a \in\mathbb{Z}_{> 0}},\operatorname{GCD} \right\rangle \). Since the latter theory was proved decidable in the first part, this reduction completes the proof of the theorem. Analogues of two lemmas of Lipshitz's proof are used in the step of variable isolation for quasi-elimination. In the quasi-elimination step we apply GCD-lemma, which was proved in the first part.
For Part I see [the author, Vestn. St. Petersbg. Univ., Math. 54, No. 3, 264--272 (2021; Zbl 1491.03011); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 8(66), No. 3, 455--466 (2021)].Number theory and combinatorics. A collection in honor of the mathematics of Ronald Grahamhttps://zbmath.org/1491.050022022-09-13T20:28:31.338867ZPublisher's description: This volume is dedicated to the work and memory of Professor Ronald L. Graham known as the architect of discrete mathematics and combinatorics and will consist of up to 20 contributions from top mathematicians reflecting on his work in combinatorics and number theory.
\begin{itemize}
\item Original contributions by leading experts in combinatorics and number theory
\item Includes essays and memories of Professor Graham from those that knew him
\item Of interest to researchers and graduate students working in combinatorics and number theory.
\end{itemize}
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Adeniran, Ayomikun; Snider, Lauren; Yan, Catherine}, Multivariate difference Gončarov polynomials, 1-20 [Zbl 07571262]
\textit{Butler, Steve (ed.); Hurlbert, Glenn (ed.)}, Foreword, V-VI [Zbl 07571285]
\textit{Allouche, J.-P.}, On an inequality in a 1970 paper of R. L. Graham, 21-26 [Zbl 07571263]
\textit{Alon, Noga; Alweiss, Ryan; Liu, Yang P.; Martinsson, Anders; Narayanan, Shyam}, Arithmetic progressions in sumsets of sparse sets, 27-33 [Zbl 07571264]
\textit{Bennett, Michael A.; Martin, Greg; O'bryant, Kevin}, Multidimensional Padé approximation of binomial functions: equalities, 35-64 [Zbl 07571265]
\textit{Blomberg, Lars; Shannon, S. R.; Sloane, N. J. A.}, Graphical enumeration and stained glass windows. I: Rectangular grids, 65-97 [Zbl 07571266]
\textit{Brown, Tom C.; Mohsenipour, Shahram}, Two extensions of Hilbert's cube lemma, 99-107 [Zbl 07571267]
\textit{Budden, Mark}, The Gallai-Ramsey number for a tree versus complete graphs, 109-113 [Zbl 07571268]
\textit{Buhler, Joe; Freiling, Chris; Graham, Ron; Kariv, Jonathan; Roche, James R.; Tiefenbruck, Mark; van Alten, Clint; Yeroshkin, Dmytro}, On Levine's notorious hat puzzle, 115-165 [Zbl 07571269]
\textit{Cooper, Joshua; Fickes, Grant}, Recurrence ranks and moment sequences, 167-186 [Zbl 07571270]
\textit{Dudek, Andrzej; Grytczuk, Jarosław; Ruciński, Andrzej}, On weak twins and up-and-down subpermutations, 187-202 [Zbl 07571271]
\textit{Farhangi, Sohail; Grytczuk, Jarosław}, Distance graphs and arithmetic progressions, 203-208 [Zbl 07571272]
\textit{Filaseta, Michael; Juillerat, Jacob}, Consecutive primes which are widely digitally delicate, 209-247 [Zbl 07571273]
\textit{Griggs, Jerrold R.}, Spanning trees and domination in hypercubes, 249-258 [Zbl 07571274]
\textit{Harborth, Heiko; Nienborg, Hauke}, Rook domination on hexagonal hexagon boards, 259-265 [Zbl 07571275]
\textit{Hindman, Neil; Strauss, Dona}, Strongly image partition regular matrices, 267-284 [Zbl 07571276]
\textit{Hopkins, Brian}, Introducing shift-constrained Rado numbers, 285-296 [Zbl 07571277]
\textit{Lichtman, Jared Duker}, Mertens' prime product formula, dissected, 297-310 [Zbl 07571278]
\textit{Nathanson, Melvyn B.}, Curious convergent series of integers with missing digits, 311-327 [Zbl 07571279]
\textit{Pomerance, Carl}, A note on Carmichael numbers in residue classes, 321-327 [Zbl 07571280]
\textit{Shkredov, I. D.; Solymosi, J.}, Tilted corners in integer grids, 329-338 [Zbl 07571281]
\textit{Alon, Noga (ed.); Brown, Tom C (ed.); Butler, Steve (ed.); Griggs, Jerrold R. (ed.); Hindman, Neil (ed.); Jungic, Veselin (ed.); Landman, Bruce M. (ed.); Nešetřil, Jaroslav (ed.)}, Remembrances [of Ron Graham], 339-360 [Zbl 07571282]
\textit{Butler, Steve}, A selected bibliography of Ron Graham, 355-360 [Zbl 07571283]On combinatorial rectangles with minimum \(\infty \)-discrepancyhttps://zbmath.org/1491.050092022-09-13T20:28:31.338867Z"Song, Chunwei"https://zbmath.org/authors/?q=ai:song.chunwei"Yao, Bowen"https://zbmath.org/authors/?q=ai:yao.bowenSummary: A combinatorial rectangle may be viewed as a matrix whose entries are all \(\pm 1\). The discrepancy of an \(m\times n\) matrix is the maximum among the absolute values of its \(m\) row sums and \(n\) column sums. In this paper, we investigate combinatorial rectangles with minimum discrepancy (0 or 1 for each line depending on the parity). Specifically, we get explicit formula for the number of matrices with minimum \(L^\infty \)-discrepancy up to 4 rows, and establish the order of magnitude of the number of such matrices with \(m\) rows and \(n\) columns while \(m\) is fixed and \(n\) approaches infinity.
Our main idea is to count column-good matrices (matrices whose column sum attains minimum discrepancy) for a given row-sum vector, and use a decreasing criterion based on a majority relation over the set of row-sum vectors. In particular, the least element of this relation, namely, row sums reaching minimum discrepancy, gives the desired count.On the maximum of the weighted binomial sum \(2^{-r}\sum_{i=0}^r\binom{m}{i}\)https://zbmath.org/1491.050102022-09-13T20:28:31.338867Z"Glasby, S. P."https://zbmath.org/authors/?q=ai:glasby.stephen-peter"Paseman, G. R."https://zbmath.org/authors/?q=ai:paseman.g-rFor a non-negative integer \(m \geq 0\), study of the function \(f_m(r) = \frac{1}{2^r} \sum_{i=0}^r \binom{m}{i}\) is of importance in coding and information theory. The authors mention that a back-of-the-envelope calculations by B. McKay in [\textit{S. P. Glasby}, ``On the maximum of the weighted binomial sum \(2^{-r}\sum_{i=0}^r \binom{m}{i}\)'', MathOverflow, Question 389857, \url{https://mathoverflow.net/questions/389857/maximum-of-the-weighted-binomial-sum-2-r-sum-i-0r-binommi}] indicate that the function has maximum value when \(r\) is close to \(m/3\). In this paper, the authors prove such a precise result and also give bounds for the maximal value. They deduce that the maximum value is asymptotic to \(\frac{3}{\sqrt{\pi m}} \bigg(\frac{3}{2} \bigg)^m\) as \(m \rightarrow \infty\). The methods are elementary.
Reviewer: Balasubramanian Sury (Bangalore)A plethora of polynomials: a toolbox for counting problemshttps://zbmath.org/1491.050122022-09-13T20:28:31.338867Z"Bogart, Tristram"https://zbmath.org/authors/?q=ai:bogart.tristram"Woods, Kevin"https://zbmath.org/authors/?q=ai:woods.kevin|woods.kevin-mSummary: A wide variety of problems in combinatorics and discrete optimization depend on counting the set \(S\) of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour through numerous problems of this type. In particular, we consider families of such sets \(S_t\) depending on one or more integer parameters \(t\), and analyze the behavior of the function \(f(t)=|S_t|\). In the examples that we investigate, this function exhibits surprising polynomial-like behavior. We end with two broad theorems detailing settings where this polynomial-like behavior must hold. The plethora of examples illustrates the framework in which this behavior occurs and also gives an intuition for many of the proofs, helping us create a toolbox for counting problems like these.A new class of generating functions of binary products of Gaussian numbers and polynomialshttps://zbmath.org/1491.050142022-09-13T20:28:31.338867Z"Boughaba, Souhila"https://zbmath.org/authors/?q=ai:boughaba.souhila"Boussayoud, Ali"https://zbmath.org/authors/?q=ai:boussayoud.ali"Kerada, Mohamed"https://zbmath.org/authors/?q=ai:kerada.mohamedSummary: In this paper, we introduce an operator in order to derive some new symmetric properties of Gaussian Fibonacci numbers and Gaussian Lucas numbers. By making use of the operator defined in this paper, we give some new generating functions for Gaussian Fibonacci numbers, Gaussian Lucas numbers, Gaussian Pell numbers, Gaussian Pell Lucas numbers, Gaussian Jacobsthal numbers, Gaussian Jacobsthal polynomials, Gaussian Jacobsthal Lucas polynomials, and Gaussian
Pell polynomials.On the combined Jacobsthal-Padovan generalized quaternionshttps://zbmath.org/1491.050172022-09-13T20:28:31.338867Z"Gürses, Nurten"https://zbmath.org/authors/?q=ai:gurses.nurten"İşbilir, Zehra"https://zbmath.org/authors/?q=ai:isbilir.zehraSummary: In this article, we examine the combined Jacobsthal-Padovan (CJP) generalized quaternions with four special cases: Jacobsthal-Padovan, Jacobsthal-Perrin, adjusted Jacobsthal-Padovan and modified Jacobsthal-Padovan generalized quaternions. Then, recurrence relation, generating function, Binet-like formula and exponential generating function of these quaternions are examined. In addition to this, some new properties, special determinant equations, matrix formulas and summation formulas are discussed.On a partition identity of Lehmerhttps://zbmath.org/1491.050212022-09-13T20:28:31.338867Z"Ballantine, Cristina"https://zbmath.org/authors/?q=ai:ballantine.cristina-m"Burson, Hannah"https://zbmath.org/authors/?q=ai:burson.hannah-e"Folsom, Amanda"https://zbmath.org/authors/?q=ai:folsom.amanda-l"Hsu, Chi-Yun"https://zbmath.org/authors/?q=ai:hsu.chi-yun"Negrini, Isabella"https://zbmath.org/authors/?q=ai:negrini.isabella"Wen, Boya"https://zbmath.org/authors/?q=ai:wen.boyaSummary: Euler's identity equates the number of partitions of any non-negative integer \(n\) into odd parts and the number of partitions of \(n\) into distinct parts. Beck conjectured and \textit{G. E. Andrews} [``Euler's partition identity and two problems of George Beck'', Math. Stud. 86, 1--2, 115--119 (2017)] proved the following companion to Euler's identity: the excess of the number of parts in all partitions of \(n\) into odd parts over the number of parts in all partitions of \(n\) into distinct parts equals the number of partitions of \(n\) with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called ``Beck-type'' companions to other identities.
In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by D. H. Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of \(n\) with an even number of even parts over the number of partitions of \(n\) with an odd number of even parts equals the number of partitions of \(n\) into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.Beck-type identities: new combinatorial proofs and a modular refinementhttps://zbmath.org/1491.050222022-09-13T20:28:31.338867Z"Ballantine, Cristina"https://zbmath.org/authors/?q=ai:ballantine.cristina-m"Welch, Amanda"https://zbmath.org/authors/?q=ai:welch.amandaSummary: Let \({\mathcal{O}}_r(n)\) be the set of \(r\)-regular partitions of \(n, {\mathcal{D}}_r(n)\) the set of partitions of \(n\) with parts repeated at most \(r-1\) times, \({\mathcal{O}}_{1,r}(n)\) the set of partitions with exactly one part (possibly repeated) divisible by \(r\), and let \({\mathcal{D}}_{1,r}(n)\) be the set of partitions in which exactly one part appears at least \(r\) times. If \(E_{r, t}(n)\) is the excess in the number of parts congruent to \(t \pmod r\) in all partitions in \({\mathcal{O}}_r(n)\) over the number of different parts appearing at least \(t\) times in all partitions in \({\mathcal{D}}_r(n)\), then \(E_{r, t}(n) = |{\mathcal{O}}_{1,r}(n)| = |{\mathcal{D}}_{1,r}(n)|\). We prove this analytically and combinatorially using a bijection due to \textit{X. Xiong} and \textit{W. J. Keith} [Ramanujan J. 49, No. 3, 555--565 (2019; Zbl 1470.11266)]. As a corollary, we obtain the first Beck-type identity, i.e., the excess in the number of parts in all partitions in \(\mathcal{O}_r(n)\) over the number of parts in all partitions in \(\mathcal{D}_r(n)\) equals \((r - 1)|\mathcal{O}_{1,r}(n)|\) and also \((r - 1)|\mathcal{D}_{1,r}(n)|\). Our work provides a new combinatorial proof of this result that does not use Glaisher's bijection. We also give a new combinatorial proof based of the Xiong-Keith bijection for a second Beck-type identity that has been proved previously using Glaisher's bijection.Corners of self-conjugate \((s,s + 1)\)-cores and \(( \overline{s}, \overline{s + 1})\)-coreshttps://zbmath.org/1491.050232022-09-13T20:28:31.338867Z"Cho, Hyunsoo"https://zbmath.org/authors/?q=ai:cho.hyunsoo"Hong, Kyounghwan"https://zbmath.org/authors/?q=ai:hong.kyounghwanSummary: In this paper, we describe a result on self-conjugate \((s, s + 1)\)-core partitions with the fixed number of corners. We also define shifted corners of a distinct partition and find formulas for the number of \(( \overline{s}, \overline{s + 1})\)-core partitions and the number of \((s, s + 1)\)-core shifted Young diagrams with the fixed number of shifted corners.A combinatorial proof of a Schmidt type theorem of Andrews and Paulehttps://zbmath.org/1491.050242022-09-13T20:28:31.338867Z"Ji, Kathy Q."https://zbmath.org/authors/?q=ai:ji.kathy-qingSchmidt partitions were introduced in [\textit{F. Schmidt}, ``Problem 10629'', Am. Math. Mon. 104, No. 10, 974 (1997)]. They are similar to integer partitions with distinct parts, but with an unusual ``size'' parameter. It is showed there that, for each integer \(n\),
\par i) the number of integer partitions of \(n\),
\par ii) the number of Schmidt partitions of \(n\), \textit{i.e.}, partitions \(\lambda_1>\lambda_2>\cdots > \lambda_k\) (for some \(k\geq 0\)) such that \(\lambda_1+\lambda_3+\lambda_5+\cdots = n\),
are equal.
\textit{G. E. Andrews} and \textit{P. Paule} [J. Number Theory 234, 95--119 (2022; Zbl 1484.11201)] showed the following variant. For each integer \(n\),
\par 1) the number of two-colored partitions of \(n\),
\par 2) the number of Schmidt partitions of \(n\), \textit{i.e.}, partitions \(\lambda_1\geq \lambda_2\geq \cdots \geq \lambda_k\) (for some \(k\geq 0\)) such that \(\lambda_1+\lambda_3+\lambda_5+\cdots = n\),
are equal.
In the present note, the authors give a very natural bijective proof of the previous result, via some tools such as 2-modular diagrams, Sylvester's bijection, and Wright's bijection. The bijection also gives a refinement of the result by Andrews and Paule [loc. cit.].
Reviewer: Matthieu Josuat-Vergès (Paris)On \(\ell \)-regular cubic partitions with odd parts overlinedhttps://zbmath.org/1491.050252022-09-13T20:28:31.338867Z"Naika, M. S. Mahadeva"https://zbmath.org/authors/?q=ai:mahadeva-naika.megadahalli-sidda"Harishkumar, T."https://zbmath.org/authors/?q=ai:harishkumar.thippeswamyLet \(\overline{a}_\ell(n)\) denote the number of \(\ell\)-regular cubic partitions of \(n\) where the first occurrence of each distinct odd part may be overlined. Thus the generating function for \(\overline{a}_\ell(n)\) is given by
\[
\sum_{n=0}^\infty\overline{a}_\ell(n)q^n=\dfrac{(-q;q^2)_\infty(q^\ell;q^\ell)_\infty(q^{2\ell};q^{2\ell})_\infty} {(q;q)_\infty(q^2;q^2)_\infty(-q^\ell;q^{2\ell})_\infty},
\]
where
\[
(a;q)_\infty=\prod_{n=0}^\infty(1-aq^n),\qquad|q|<1.
\]
Utilizing some standard \(q\)-series techniques, the authors establish many infinite families of congruences modulo powers of \(2\) and \(3\) for \(\overline{a}_3(n)\) and modulo powers of \(2\) for \(\overline{a}_5(n)\). For example, they prove that for any \(n\geq0\),
\begin{align*}
\overline{a}_3{\left(2^{2\alpha+5}\cdot5^{2\beta+2}n+\dfrac{a\times2^{2\alpha+4}\cdot5^{2\beta+1}-1}{3}\right)} &\equiv0\pmod{27},\\
\overline{a}_5{\left(2^3\cdot5^{2\beta+1}n+\dfrac{b\times5^{2\beta}-2}{3}\right)} &\equiv0\pmod{32},
\end{align*}
where \(a\in\{11,17,23,29\}\) and \(b\in\{23,47\}\). The main ingredient in proofs are some 2- and 3-dissections identities for certain \(q\)-series.
Reviewer: Dazhao Tang (Chongqing)Congruences for \((s, t)\)-regular bipartition quadrupleshttps://zbmath.org/1491.050262022-09-13T20:28:31.338867Z"Nayaka, S. Shivaprasada"https://zbmath.org/authors/?q=ai:nayaka.s-shivaprasadaSummary: Let \(BQ_{s, t}(n)\) denote the number of \((s, t)\)-regular bipartition quadruples of a positive integer \(n\). We obtain some Ramanujan-type congruences for \(BQ_{s, t}(n)\) modulo 5, 128 and 512 for various values of \(s\) and \(t\). For examples,
\[
\begin{aligned}&BQ_{2, 2}(8n+7)\equiv 0 \pmod{512},\\
&BQ_{5, 5}\left( 25\cdot 4^{\alpha +2}n+\frac{250\cdot 4^{\alpha +1}-8}{3}\right) \equiv 0 \pmod{5} \end{aligned}
\]
for all nonnegative integers \(\alpha\) and \(n\).Combinatorial identities involving reciprocals of the binomial and central binomial coefficients and harmonic numbershttps://zbmath.org/1491.050282022-09-13T20:28:31.338867Z"Batır, Necdet"https://zbmath.org/authors/?q=ai:batir.necdet"Chen, Kwang-Wu"https://zbmath.org/authors/?q=ai:chen.kwang-wuThe authors prove a general combinatorial formula involving the reciprocals of the binomial coefficients and the partial sums of an arbitrary sequence. The proof uses induction and a recurrence formula. Applications of this formula include several combinatorial identities involving reciprocals of the binomial and central binomial coefficients and harmonic numbers. The oldest examples to which the main result is applicable were already published in the textbook of \textit{R. L. Graham} et al. [Concrete mathematics: a foundation for computer science. 2nd ed. Amsterdam: Addison-Wesley Publishing Group (1994; Zbl 0836.00001)], most existing examples appeared in the work of \textit{R. Wituła} and \textit{D. Słota} [Asian-Eur. J. Math. 1, No. 3, 439--448 (2008; Zbl 1168.05005)]. The authors also provide some new applications.
Reviewer: Gabor Hetyei (Charlotte)Inverse relations and reciprocity laws involving partial Bell polynomials and related extensionshttps://zbmath.org/1491.050302022-09-13T20:28:31.338867Z"Schreiber, Alfred"https://zbmath.org/authors/?q=ai:schreiber.alfredSummary: The objective of this paper is, mainly, twofold: Firstly, to develop an algebraic setting for dealing
with Bell polynomials and related extensions. Secondly, based on the author's previous work on multivariate
Stirling polynomials [Discrete Math. 338, No. 12, 2462--2484 (2015; Zbl 1321.11030)], to present a number of new results related to different types of inverse relationships, among these (1) the use of multivariable Lah polynomials for characterizing self-orthogonal families of
polynomials that can be represented by Bell polynomials, (2) the introduction of `generalized Lagrange inversion polynomials' that invert functions characterized in a specific way by sequences of constants, (3) a general
reciprocity theorem according to which, in particular, the partial Bell polynomials \(B_{n,k}\) and their orthogonal
companions \(A_{n,k}\) belong to one single class of Stirling polynomials: \(A_{n,k} = (-1)^{n-k}B_{-k,-n}\). Moreover, of some
numerical statements (such as Stirling inversion, Schlömilch-Schläfli formulas) generalized polynomial versions
are established. A number of well-known theorems (Jabotinsky, Mullin-Rota, Melzak, Comtet) are given new
proofs.Laguerre inequalities for discrete sequenceshttps://zbmath.org/1491.050312022-09-13T20:28:31.338867Z"Wang, Larry X. W."https://zbmath.org/authors/?q=ai:wang.larry-x-w"Yang, Eve Y. Y."https://zbmath.org/authors/?q=ai:yang.eve-y-ySummary: The Turán inequality, the Laguerre inequality and their \(m\)-rd generalizations have been proved to be closely relative with the Laguerre-Pólya class and Riemann hypothesis. Since these two inequalities are equivalent to log-concavity of the discrete sequences, we consider whether their generalizations hold for discrete sequences. Recently, \textit{W. Y. C. Chen} et al. [Trans. Am. Math. Soc. 372, No. 3, 2143--2165 (2019; Zbl 1415.05020)] proved that the partition function satisfies the Turán inequality of order 2 and thus the 3-rd Jensen polynomials associated with the partition function have only real zeros. \textit{M. Griffin} et al. [Proc. Natl. Acad. Sci. USA 116, No. 23, 11103--11110 (2019; Zbl 1431.11105)] proved the \(n\)-th Jensen polynomials associated with the Maclaurin coefficients of the function in the Laguerre-Pólya class and the partition function have only real zeros except finite terms. In this paper, we show the Laguerre inequality of order 2 is true for the partition function, the overpartition function, the Bernoulli numbers, the derangement numbers, the Motzkin numbers, the Fine numbers, the Franel numbers and the Domb numbers.Heine's transformation formula through \(q\)-difference equationshttps://zbmath.org/1491.050322022-09-13T20:28:31.338867Z"Arjika, Sama"https://zbmath.org/authors/?q=ai:arjika.sama"Chaudhary, M. P."https://zbmath.org/authors/?q=ai:chaudhary.mahendra-pal"Hounkonnou, M. N."https://zbmath.org/authors/?q=ai:hounkonnou.mahouton-norbertSummary: In this paper, we give an extension of the first Heine's transformation formula using \(q\)-difference equations. Further, we discussed a Ramanujan's theta function \(\psi(q)\) and deduced it as a particular case.On radiuses of convergence of \(q\)-metallic numbers and related \(q\)-rational numbershttps://zbmath.org/1491.050342022-09-13T20:28:31.338867Z"Ren, Xin"https://zbmath.org/authors/?q=ai:ren.xinSummary: The \(q\)-rational numbers and the \(q\)-irrational numbers were introduced by \textit{S. Morier-Genoud} and \textit{V. Ovsienko} [Forum Math. Sigma 8, Paper No. e13, 55 p. (2020; Zbl 1434.05023)]. In this paper, we focus on \(q\)-real quadratic irrational numbers, especially \(q\)-metallic numbers and \(q\)-rational sequences which converge to \(q\)-metallic numbers, and consider the radiuses of convergence of them when we assume that \(q\) is a complex number. We construct two sequences given by recurrence formula as a generalization of the \(q\)-deformation of the Fibonacci and Pell numbers which are introduced by Morier-Genoud and Ovsienko [loc. cit.]. For these two sequences, we prove a conjecture of \textit{L. Leclere} et al. [``On radius of convergence of \(q\)-deformed real numbers'', Preprint, \url{arXiv:2102.00891}] concerning the expected lower bound of the radiuses of convergence. In addition, we obtain a relationship between the radius of convergence of these two sequences in two special cases.Representations of degenerate Hermite polynomialshttps://zbmath.org/1491.050362022-09-13T20:28:31.338867Z"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san"Jang, Lee-Chae"https://zbmath.org/authors/?q=ai:jang.lee-chae|jang.leechae"Lee, Hyunseok"https://zbmath.org/authors/?q=ai:lee.hyunseok"Kim, Hanyoung"https://zbmath.org/authors/?q=ai:kim.hanyoungSummary: The study on degenerate versions of some special numbers and polynomials, which began with Carlitz's pioneering work, has regained recent interests of some mathematicians. Motivated by this, we introduce degenerate Hermite polynomials as a degenerate version of the ordinary Hermite polynomials. Recently, introduced was \(\lambda\)-umbral calculus where the usual exponential function appearing in the generating function of Sheffer sequence is replaced by the degenerate exponential function. Then, among other things, by using the formula of the \(\lambda\)-umbral calculus about expressing one \(\lambda\)-Sheffer polynomial in terms of another \(\lambda\)-Sheffer polynomials we represent the degenerate Hermite polynomials in terms of the higher-order degenerate Bernoulli, Euler, and Frobenius-Euler polynomials and vice versa.Weyl orbits of matrix morsifications and a Coxeter spectral classification of positive signed graphs and quasi-Cartan matrices of Dynkin type \(\mathbb{A}_n\)https://zbmath.org/1491.051002022-09-13T20:28:31.338867Z"Simson, Daniel"https://zbmath.org/authors/?q=ai:simson.danielSummary: By applying an advanced technique of Weyl orbits of matrix morsifications of Dynkin graphs we obtain a complete classification (up to the strong Gram \(\mathbb{Z}\)-congruence) of the connected positive signed graphs \(\Delta\), with \(n \geq 2\) vertices, of Dynkin type \(\mathbb{A}_n\) (see Problem 2.1(a)), by a reduction of the problem to the combinatorial properties of the Weyl group of type \(\mathbb{A}_n\) and to the classification of the conjugacy classes of the integer orthogonal matrices \(C\) in \(\mathbb{M}_{n + 1}(\mathbb{Z})\) such that the characteristic polynomial of \(C\) is of the form \(\operatorname{char}_C(t) = t^{n + 1} - 1\). Following the usual Coxeter spectral analysis technique of edge-bipartite graphs applied in \textit{D. Simson} [Linear Algebra Appl. 557, 105--133 (2018; Zbl 1396.05049)], we study such positive signed graphs \(\Delta\), with \(n \geq 2\) vertices, by means of the non-symmetric Gram matrix \(\check{G}_{\Delta} \in \mathbb{M}_n(\mathbb{Z})\) defining \(\Delta\), its Gram quadratic form \(q_{\Delta} : \mathbb{Z}^n \to \mathbb{Z}\), \(v \mapsto v \cdot \check{G}_{\Delta} \cdot v^{t r}\), the complex spectrum \(\mathrm{specc}_{\Delta} \subset \mathcal{S}^1 : = \{z \in \mathbb{C}, | z | = 1 \}\) of the Coxeter matrix \(\mathrm{Cox}_{\Delta} : = - \check{G}_{\Delta} \cdot \check{G}_{\Delta}^{- t r} \in \mathbb{M}_n(\mathbb{Z})\), called the Coxeter spectrum of \(\Delta\), and the Coxeter polynomial \(\mathrm{cox}_{\Delta}(t) : = \det(t \cdot E - \mathrm{Cox}_{\Delta}) \in \mathbb{Z} [t]\). We define \(\Delta\) to be positive if the quadratic form \(q_{\Delta} : \mathbb{Z}^n \to \mathbb{Z}\) is positive definite, or equivalently the symmetric Gram matrix \(G_{\Delta} : = \frac{ 1}{ 2} [ \check{G}_{\Delta} + \check{G}_{\Delta}^{tr}] = E + \frac{ 1}{ 2} \operatorname{Ad}_{\Delta} \in \mathbb{M}_n(\frac{ 1}{ 2} \mathbb{Z})\) of \(\Delta\) is positive definite, where \(\operatorname{Ad}_{\Delta} \in \mathbb{M}_n(\mathbb{Z})\) is the adjacency matrix of \(\Delta\).
Here we classify such positive signed graphs, up to the strong Gram \(\mathbb{Z}\)-congruence \(\Delta \approx_{\mathbb{Z}} \Delta^\prime\), where \(\Delta \approx_{\mathbb{Z}} \Delta^\prime\) means that \(\check{G}_{\Delta^\prime} = B^{tr} \cdot \check{G}_{\Delta} \cdot B\), for some \(B \in \mathbb{M}_n(\mathbb{Z})\) with \(\det B = \pm 1\). The main result of the paper asserts that, given such a connected positive signed graph \(\Delta\) of Dynkin type \(\mathbb{A}_n\) (equivalently, \(\det 2 G_{\Delta} = n + 1)\), we have
\par (i) the Coxeter polynomial \(\mathrm{cox}_{\Delta}(t)\) has the form \(t^n + t^{n - 1} + \cdots + t + 1\) (i.e., it coincides with the Coxeter polynomial \(\mathrm{cox}_{\mathbb{A}_n}(t)\) of the Dynkin graph \(\mathbb{A}_n)\), and
\par (ii) there is a strong Gram \(\mathbb{Z}\)-congruence \(\Delta \approx_{\mathbb{Z}} \mathbb{A}_n\).
As a consequence we obtain the following solution of the Coxeter spectral hypothesis stated in [\textit{D. Simson}, SIAM J. Discrete Math. 27, No. 2, 827--854 (2013; Zbl 1272.05072); Linear Algebra Appl. 586, 190--238 (2020; Zbl 1429.05133)]:
\par (a) Given a pair \(\Delta\), \(\Delta^\prime\) of connected positive signed graphs of Dynkin type \(\mathbb{A}_n\), with \(n \geq 2\) vertices, there is a congruence \(\Delta \approx_{\mathbb{Z}} \Delta^\prime\) if and only if \(\mathbf{specc}_{\Delta} = \mathrm{specc}_{\Delta^\prime}\), and
\par (b) Given a positive definite connected quasi-Cartan matrix \(C \in \mathbb{M}_n(\mathbb{Z})\) (in the sense of [\textit{M. Barot} et al., J. Lond. Math. Soc., II. Ser. 73, No. 3, 545--564 (2006; Zbl 1093.05070)]) of Dynkin type \(\mathbb{A}_n\) (equivalently, \(\det C = n + 1)\), the Coxeter polynomial \(\mathrm{cox}_C(t) \in \mathbb{Z} [t]\) of \(C\) coincides with the Coxeter polynomial \(\mathrm{cox}_{\mathbb{A}_n}(t) = t^n + t^{n - 1} + \cdots + t + 1\) of the Dynkin graph \(\mathbb{A}_n\) and \(C\) is strongly \(\mathbb{Z}\)-congruent with the canonical symmetrizable Cartan matrix \(\mathbb{A}_n\) of the root system of type \(\mathbb{A}_n\).Numbers. Arithmetic and computationhttps://zbmath.org/1491.110012022-09-13T20:28:31.338867Z"Mallik, Asok Kumar"https://zbmath.org/authors/?q=ai:mallik.asok-kumar"Das, Amit Kumar"https://zbmath.org/authors/?q=ai:das.amit-kumarPublisher's description: This book contains a number of elementary ideas on numbers, their representations, interesting arithmetical problems and their analytical solutions, fundamentals of computers and programming plus programming solutions as an alternative to the analytical solutions and much more.
Spanning seven chapters, this book, while keeping its lucid storytelling verve, describes integers, real numbers and numerous interesting properties and historical references; followed by a good collection of arithmetic problems and their analytical solutions.Continued fractions and signal processinghttps://zbmath.org/1491.110022022-09-13T20:28:31.338867Z"Sauer, Tomas"https://zbmath.org/authors/?q=ai:sauer.tomasThe book gives an extensive overview of the concept ``continued fractions''. For the history the author refers to [\textit{O. Becket}, ``Quellen und Studien zur Geschichte'', Math. Astron. Physik B2, 311--333 (1933)].
The book begins with a nicely written preface with subsections prerequisites and recommended reading, literature and continued fractions and a section called personal remarks, presenting the manner in which the author indicates how he met the subject and what triggered him to write a book.
The contents of the book are mainly on the subject of continued fractions (six of the seven chapters); there is only one chapter (40 pages) about the subject of signal analysis; the concepts mentioned are only `touched upon'. In the list of References the customary references to publications on continued fractions appear (Euler, Gauss, Wall, Perron, Khinchine).
Moreover, every section in every chapter ends with several problems which lead to a better understanding of the ground covered.
For completeness, a short overview of the chapters and their contents is given below.
Chapter 1. Continued fractions and what can be done with them.
Chapter 2. Continued fractions of real numbers.
Chapter 3. Rational functions as continued fractions of polynomials.
Chapter 4. Continued fractions and Gauss.
Chapter 5. Continued fractions and Prony.
Chapter 6. Digital signal processing.
Subsections signals and filters, Fourier and sampling, realization of filters, rational filters and stability, stability of difference equations, superresolution via continued fractions and a determinantal identity.
Chapter 7. Continued fractions, Hurwitz and Stieltjes.
Finally a list of references (\(114\) items) is given, along with a six page index to facilitate browsing the book.
Reviewer: Marcel G. de Bruin (Heemstede)Towards a dimension formula for automorphic forms on \(O(II_{2,10})\)https://zbmath.org/1491.110032022-09-13T20:28:31.338867Z"Rössler, Maximilian"https://zbmath.org/authors/?q=ai:rossler.maximilianSummary: This thesis is concerned with the computation of dimension formulas for special orthogonal modular forms associated with the \(II_{2,10}\)-lattice. For a given arithmetic group, the dimension of the spaces of these orthogonal modular forms is a polynomial of degree 10 in the weight. By using the Hirzebruch-Riemann-Roch theorem and Hirzebruch-Mumford proportionality, this polynomial can be determined up to a geometric error term; this error term is a linear polynomial whose coefficients are given by intersection products of toroidal boundary divisors and certain logarithmic Chern classes. We describe this error term in more detail and determine important components. For this purpose, we construct a special toroidal compactification of the orthogonal moduli variety associated to the \(II_{2,10}(N)\)-lattice and study its geometry. We also describe an essential part of the intersection theory of this compactification, thus reducing the computation of the linear coefficient of the error term to a combinatorial problem. Finally, we give methods to reduce the computation of the constant coefficient of the error term to combinatorial problems; in particular, we can formulate a formulation of the error term without logarithmic Chern classes.The Bachet-Bézout theoremhttps://zbmath.org/1491.110042022-09-13T20:28:31.338867Z"Hauchecorne, Bertrand"https://zbmath.org/authors/?q=ai:hauchecorne.bertrandSummary: Le théorème (ou l'identité) de Bézout s'énonce classiquement soit pour les entiers, soit pour les polynômes, mais de façon plus générale, il affirme que deux éléments \(a\) et \(b\) d'un anneau principal \(A\) sont premiers entre eux si et seulement s'il existe deux elements \(u\) et \(v\) de \(A\) tel que \(ua+bv-1\). C'est par un long cheminement partant d'Euclide, passant par le Sieur de Méziriac puis par Etienne Bézout que l'on y est parvenu. La réciproque en est évidente, puisque si \(a\) et \(b\) ont un diviseur commun, celui-ci divise \(ua+vb\) donc 1, ce qui prouve que 1 est le seul diviseur commun et qu'en conséquence \(a\) et \(b\) sont premier entre eux. C'est donc le sens direct qui nous interesse.InfoMod: a visual and computational approach to Gauss' binary quadratic formshttps://zbmath.org/1491.110052022-09-13T20:28:31.338867Z"Zeytin, Ayberk"https://zbmath.org/authors/?q=ai:zeytin.ayberkSummary: InfoMod is a software devoted to the modular group, \(\text{PSL}_2 (\mathbb{Z})\). It consists of algorithms that deal with the classical correspondences among geodesics on the modular surface, elements of the modular group and binary quadratic forms. In addition, the software implements the recently discovered representation of Gauss' indefinite binary quadratic forms and their classes in terms of certain infinite planar graphs (dessins) called çarks. InfoMod illustrates various aspects of these forms, i.e. Gauss' reduction algorithm, the representation problem of forms, ambiguous and reciprocal forms. It can be used for visualization, for high performance computation involving these mathematical structures as well as experimenting.The mathematical artist. A tribute to John Horton Conwayhttps://zbmath.org/1491.110062022-09-13T20:28:31.338867ZPublisher's description: This book brings together the impact of Prof. John Horton Conway, the playful and legendary mathematician's wide range of contributions in science which includes research areas -- Game of Life in cellular automata, theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. It contains transcripts where some eminent scientists have shared their first-hand experience of interacting with Conway, as well as some invited research articles from the experts focusing on Game of Life, cellular automata, and the diverse research directions that started with Conway's Game of Life. The book paints a portrait of Conway's research life and philosophical direction in mathematics and is of interest to whoever wants to explore his contribution to the history and philosophy of mathematics and computer science. It is designed as a small tribute to Prof. Conway whom we lost on April 11, 2020.
The articles of this volume will be reviewed individually.Ternary arithmetic, factorization, and the class number one problemhttps://zbmath.org/1491.110072022-09-13T20:28:31.338867Z"Bingham, Aram"https://zbmath.org/authors/?q=ai:bingham.aramSummary: Ordinary multiplication of natural numbers can be generalized to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of `3-primality' -- primality with respect to ternary multiplication -- is defined, and it turns out that there are very few 3-primes. They correspond to imaginary quadratic fields \(\mathbb{Q}(\sqrt{-n})\), \(n> 0\), with odd discriminant and whose ring of integers admits unique factorization. We also describe how to determine representations of numbers as ternary products and related algorithms for usual primality testing and integer factorization.Directed graphs from exact covering systemshttps://zbmath.org/1491.110082022-09-13T20:28:31.338867Z"Neidmann, Dana"https://zbmath.org/authors/?q=ai:neidmann.danaThe paper introduces a family of directed graphs on the set of integers associated with the covering system: Given an exact covering system \(S=\{x\equiv a_i \pmod {d_i} : 1 \le i \le r\}\), in which each integer \(n\) satisfies \(n \equiv a_i \pmod {d_i}\) for exactly one value of \(i\), create the corresponding exact covering system digraph \(G_S=G(d_1n+a_1,\ldots, d_rn+a_r):=(V,E) \), where \(V(G_S)=\mathbb{Z} \) and \(E(G_S)=\{(n, d_in+a_i) : 1 \le i \le r\} \); that is, the vertices of \(G_S\) are the integers and the edges are \( (n, d_in+a_i)\) for each \(n\in\mathbb{Z}\) and for each congruence in the covering system. The author studies the structural properties of the exact covering system digraphs which have finitely many components, one cycle per component, indegree \(1\) and outdegree \(r\) at each vertex. Special attention is paid to the structure of digraphs of degree \(1\) and of degree at least \(2\). The author also finds graph isomorphisms between different exact covering system digraphs. The paper explores the link between digraphs that have a single component and non-standard digital representation of integers.
Reviewer: Armen Bagdasaryan (Madīnat al-Kuwait)On the Lévy constants of Sturmian continued fractionshttps://zbmath.org/1491.110092022-09-13T20:28:31.338867Z"Bugeaud, Yann"https://zbmath.org/authors/?q=ai:bugeaud.yann"Kim, Dong Han"https://zbmath.org/authors/?q=ai:kim.donghan.1"Lee, Seul Bee"https://zbmath.org/authors/?q=ai:lee.seul-beeLet \(P_n(\alpha)/Q_n(\alpha)\) be the \(n\)-th convergent [here denoted as ``partial quotient''] of a real irrational number \(\alpha\). Its Lévy constant is given by the following limit, if it exists and is finite \[ \mathcal{L}(\alpha) = \lim_{n \rightarrow \infty}\frac{1}{n} \log Q_n(\alpha). \]
The authors recall classical and more recent properties of the Lévy constant. Almost all real numbers have a Lévy constant equal to \(\pi^2/(12 \log 2)\); the set of irrational numbers which do not have a finite Lévy constant has a full Hausdorff dimension. Lagrange proved that an irrational \(\alpha\) is quadratic if and only if the sequence of its Farey coefficients is ultimately periodic and more recently the Lévy constant of a quadratic number has been explicitly given in terms of its coefficients.
The author consider the ``simplest'' \(\alpha\) for which the sequence of its coefficients is bounded and not ultimately periodic. One analyses the ``complexity'' of a sequence \(\mathbf{a}=(a_i)_i\) of bounded integers by the number of its blocks of length \(n\), denoted by \(p_{\mathbf{a}}(n)\). If the sequence \(\mathbf{a}\) is not ultimately periodic, one has \(p_{\mathbf{a}}(n) \ge n+1\) and a sequence \(\mathbf{a}\) satisfying \(p_{\mathbf{a}}(n) = n+1\) is called Sturmian. Those sequences have been much studied and a property due to \textit{V. Berthé} [Theor. Comput. Sci. 165, No. 2, 295--309 (1996; Zbl 0872.11018)], obtained by dynamical system considerations, implies that a number \(\alpha\) having a Sturmian sequence of Farey coefficients has a finite Lévy constant.
In Section 2, the authors give a purely combinatorial proof of this last result, and they also extend it to ``quasi-Sturmian'' \(\alpha\) satisfying \(p_{\mathbf{a}}(n) \le n+k\), considered by Cassaigne.
The main result of the paper, also obtained by combinatorial arguments, states a slightly more general result that what we state here for simplicity
Theorem [1.2]. Let \(a, b\) be integers with \(1 \le a < b \). The set of Lévy constants of Sturmian irrationals with Farey coefficients in \(\{a, b\}\) is equal to the whole interval \([\mathcal{L}([0;\bar{a}], [\mathcal{L}([0;\bar{b}]]\).
This easily implies that the set of Lévy constants of irrational real numbers the sequence of coefficients of which is periodic or Sturmian is the whole interval \([\log((1+\sqrt{5})/2), +\infty)\).
A key object of the study is the quantity \[ T(a_1, \ldots,a_n) = \operatorname{Tr}\left(\begin{pmatrix} a_1 & 1 \\
1 & 0 \end{pmatrix} \cdots \begin{pmatrix} a_n & 1 \\
1 & 0 \end{pmatrix} \right) \] used by \textit{H. Jager} and \textit{P. Liardet} [Indag. Math. 50, No. 2, 181--197 (1988; Zbl 0655.10045)] for giving an explicit value of \(\mathcal{L}(\alpha)\) when \(\alpha\) is quadratic.
Reviewer: Jean-Marc Deshouillers (Bordeaux)On the smallest base in which a number has a unique expansionhttps://zbmath.org/1491.110102022-09-13T20:28:31.338867Z"Allaart, Pieter"https://zbmath.org/authors/?q=ai:allaart.pieter-c"Kong, Derong"https://zbmath.org/authors/?q=ai:kong.derongThis paper is concerned with the expansion of a real number \(x\in[0,1/(q-1)]\) in base \(q\in(1,2]\), called \(q\)-expansion. That is, we write
\[
x=\sum_{i\ge1}\frac{d_i}{q^i}, \quad\text{where } d_i\in\{0,1\}\quad\text{for all } i\geq 1.
\]
Such expansions are usually highly non-unique: for each \(q\in(1,2)\), almost all \(x\in[0,1/(q-1)]\) have continuum many \(q\)-expansions.
What is the infimum of the set of bases \(q\) such that \(x\) has a unique \(q\)-expansion? The central quantities in the paper under review are
\[
q_s(x):=\inf \bigl\{ q\in(1,2]: x\mbox{ has a unique \(q\)-expansion} \bigr\}
\]
and
\[
L(q):=\bigl\{x>0:q_s(x)=q\bigr\}.
\]
A number of results is proved for these quantities, including questions on cardinality, continuity, accumulation points, extreme values, monotonicity, the description and investigation of an algorithm for computing \(q_s(x)\), and a connection to de Vries-Komornik numbers. Moreover, the cases where the infimum in the definition of \(q_s(x)\) is in fact a minimum is investigated closely. Among other things, certain subsets of the the graph of \(q_s\) are studied, so-called \textit{Komornik-Loreti cascades}.
Reviewer: Lukas Spiegelhofer (Wien)On the Zeckendorf representation of smooth numbershttps://zbmath.org/1491.110112022-09-13T20:28:31.338867Z"Bugeaud, Yann"https://zbmath.org/authors/?q=ai:bugeaud.yannThe author establishes various relations between the number of digits in the Zeckendorf representation and the prime factor decomposition of integers. In particular, a large number that is a sum of few Fibonacci numbers cannot have only small prime factors. The author considers both the greatest prime factor and the quantity given by dividing a number by all its prime factors that are not in a fixed finite set. The proofs use estimates for linear forms in complex logarithms of algebraic numbers, and the involved constants are thus effectively computable.
Reviewer: Wolfgang Steiner (Paris)Exponential lower bounds on the generalized Erdős-Ginzburg-Ziv constanthttps://zbmath.org/1491.110122022-09-13T20:28:31.338867Z"Bitz, Jared"https://zbmath.org/authors/?q=ai:bitz.jared"Griffith, Sarah"https://zbmath.org/authors/?q=ai:griffith.sarah"He, Xiaoyu"https://zbmath.org/authors/?q=ai:he.xiaoyuSummary: For a finite abelian group \(G\), the generalized Erdős-Ginzburg-Ziv constant \(\mathsf{s}_k(G)\) is the smallest \(m\) such that a sequence of \(m\) elements in \(G\) always contains a \(k\)-element subsequence which sums to zero. If \(n=\exp(G)\) is the exponent of \(G\), the previously best known bounds for \(\mathsf{s}_{kn}(C_n^r)\) were linear in \(n\) and \(r\) when \(k\geq 2\). Via a probabilistic argument, we produce the exponential lower bound
\[
\mathsf{s}_{2n}(C_n^r) > \frac{n}{2}[1.25+o(1)]^r
\]
for \(n > 0\). For the general case, we show
\[
\mathsf{s}_{kn}(C_n^r) > \frac{kn}{4}(1+\frac{1}{ek+1}+o(1))^r.
\]Green's problem on additive complements of the squareshttps://zbmath.org/1491.110132022-09-13T20:28:31.338867Z"Ding, Yuchen"https://zbmath.org/authors/?q=ai:ding.yuchenSummary: Let \(A\) and \(B\) be two subsets of the nonnegative integers. We call \(A\) and \(B\) additive complements if all sufficiently large integers \(n\) can be written as \(a+b\), where \(a\in A\) and \(b\in B\). Let \(S=\{1^2,2^2,3^2,\cdots\}\) be the set of all square numbers. Ben Green was interested in the additive complement of \(S\). He asked whether there is an additive complement \(B=\{b_n\}_{n=1}^{\infty}\subseteq\mathbb{N}\) which satisfies \(b_n=\frac{\pi^2}{16}n^2+o(n^2)\). Recently, \textit{Y.-G. Chen} and \textit{J.-H. Fang} [J. Number Theory 180, 410--422 (201; Zbl 1421.11014)] proved that if \(B\) is such an additive complement, then
\[
\limsup_{n\rightarrow\infty}\,\frac{\frac{\pi^2}{16}n^2-b_n}{n^{1/2}\log n}\geq\sqrt{\frac{2}{\pi}}\frac{1}{\log 4}.
\]
They further conjectured that
\[
\limsup_{n\rightarrow\infty}\,\frac{\frac{\pi^2}{16}n^2-b_n}{n^{1/2}\log n}=+\infty.
\]
In this paper, we confirm this conjecture by giving a much more stronger result, i.e.,
\[
\limsup_{n\rightarrow\infty}\,\frac{\frac{\pi^2}{16}n^2-b_n}{n}\geq\frac{\pi }{4}.
\]On generalized perfect difference sumsetshttps://zbmath.org/1491.110142022-09-13T20:28:31.338867Z"Fang, Jin-Hui"https://zbmath.org/authors/?q=ai:fang.jinhuiA variant of the proof of van der Waerden's theorem by Furstenberghttps://zbmath.org/1491.110152022-09-13T20:28:31.338867Z"Eyidoğan, Sadık"https://zbmath.org/authors/?q=ai:eyidogan.sadik"Özkurt, Ali Arslan"https://zbmath.org/authors/?q=ai:ozkurt.ali-arslanSummary: Let \(R\) be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of \(R\), we will define a metric on \(R\) using them. Then, we will use this kind of metric to obtain a variant of the proof of van der Waerden's theorem by \textit{H. Furstenberg} [Recurrence in ergodic theory and combinatorial number theory. Princeton, New Jersey: Princeton University Press (1981; Zbl 0459.28023)].On arithmetic sums of Ahlfors-regular setshttps://zbmath.org/1491.110162022-09-13T20:28:31.338867Z"Orponen, Tuomas"https://zbmath.org/authors/?q=ai:orponen.tuomasLet \(A,B\subseteq\mathbb{R}\) be Ahlfors-regular sets with Hausdorff dimension \(\alpha\) resp. \(\beta\). The authors proves that the arithmetic sum \(A+\lambda B\) has Hausdorff dimension greater or eqaul to \(\alpha + \beta (1-\alpha)/(2-\alpha)\) outside a set of parameter values \(\lambda\) of Hausdorff dimension zero. This inserting result on Ahlfors-regular sets is stronger than the result on lower bounds for the dimension of arithmetic sums of arbitrary subsets of \(\mathbb{R}\) found in [\textit{J. Bourgain}, J. Anal. Math. 112, 193--236 (2010; Zbl 1234.11012)].
Reviewer: Jörg Neunhäuserer (Goslar)Non-commutative methods in additive combinatorics and number theoryhttps://zbmath.org/1491.110172022-09-13T20:28:31.338867Z"Shkredov, Ilya D."https://zbmath.org/authors/?q=ai:shkredov.ilya-dThis work is a survey on the area of arithmetic combinatorics, focussing on non-abelian results. Typically, additive combinatorics have dealt with problems both on the integers on abelian groups, but more recently there have been important breakthroughs on the study of questions in the non-abelian setting. For instance, the growth of sets on non-abelian groups has been a very recent trend of research with several breakthroughs on the last years.
This survey explore a wide variety of results on these area, and relates it with the abelian analogues. These includes problems on arithmetic combinatorics on itself (see for instance Section 4: The structure of sets with small doubling in an arbitrary group), as well as applications on other domains including incidence geometry, group theory and analytic number theory, among other.
Apart from a very detailed survey of the techniques (non-Fourier analysis, Balogh-Szeméredi-Gowers Theorem, etc), the author provides a very rich source of bibliography.
Reviewer: Juanjo Rué Perna (Barcelona)Linear recurrences of order at most two in small divisorshttps://zbmath.org/1491.110182022-09-13T20:28:31.338867Z"Chentouf, A. Anas"https://zbmath.org/authors/?q=ai:chentouf.a-anasIn this paper, small divisors of a positive integer \(n\) are divisors less than or equal to \(\sqrt{n}\). The author gives a complete characterization of numbers whose small divisors \(d_1 = 1 < d_2 < \ldots < d_{k}\) form a linear recurrence of order at most 2. In other words, for such numbers, there exist two integers \(a,b\) such that \(d_{i} = a d_{i-1} + b d_{i-2}\) for \(3 \leq i \leq k\). The method makes use of a tree representation of the sequences of small divisors. It turns out that all solutions with \(k \geq 5\) fall into one of the following cases:
\begin{itemize}
\item[1.] The small divisors are in geometric progression (\(b=0\));
\item[2.] The small divisors whose ranks have same parity are in geometric progression (\(a=0\));
\item[3.] The small divisors are in arithmetic progression (\(a=2\) and \(b=-1\)).
\end{itemize}
\textit{D. E. Iannucci} [Integers 18, Paper A77, 10 p. (2018; Zbl 1453.11016)] previously resolved the last case and found that \(n=60\) is the only solution of that kind with \(k \geq 5\).
Reviewer: Olivier Rozier (Paris)On a new generalization of Jacobsthal hybrid numbershttps://zbmath.org/1491.110192022-09-13T20:28:31.338867Z"Bród, D."https://zbmath.org/authors/?q=ai:brod.dorota"Szynal-Liana, A."https://zbmath.org/authors/?q=ai:szynal-liana.anettaSummary: We define a two-parameter generalization of Jacobsthal hybrid numbers. We give Binet formula, the generating functions and some identities for these numbers.More identities for Fibonacci and Lucas quaternionshttps://zbmath.org/1491.110202022-09-13T20:28:31.338867Z"Irmak, Nurettin"https://zbmath.org/authors/?q=ai:irmak.nurettinSummary: In this paper, we define the associate matrix as
\[
F= \left( \begin{matrix} 1+i+2j+3k & i+j+2k \\ i+j+2k & 1+j+k \end{matrix} \right).
\]
By the means of the matrix \(F\), we give several identities about Fibonacci and Lucas quaternions by matrix methods. Since there are two different determinant definitions of a quaternion square matrix (whose entries are quaternions), we obtain different Cassini identities for Fibonacci and Lucas quaternions apart from Cassini identities given in the papers [\textit{S. Halici}, Adv. Appl. Clifford Algebr. 22, No. 2, 321--327 (2012; Zbl 1329.11016)] and [\textit{M. Akyiğit} et al., 24, No. 3, 631--641 (2014; Zbl 1321.11020)].Products involving reciprocals of gibonacci polynomialshttps://zbmath.org/1491.110212022-09-13T20:28:31.338867Z"Koshy, Thomas"https://zbmath.org/authors/?q=ai:koshy.thomasSummary: We explore finite and infinite products involving reciprocals of gibonacci polynomials, and their Pell counterparts.On Gauss Fibonacci polynomials, on Gauss Lucas polynomials and their applicationshttps://zbmath.org/1491.110222022-09-13T20:28:31.338867Z"Özkan, Engin"https://zbmath.org/authors/?q=ai:ozkan.engin"Taştan, Merve"https://zbmath.org/authors/?q=ai:tastan.merveSummary: We define the Gauss Fibonacci polynomials. Then we give a formula for the Gauss Fibonacci polynomials by using the Fibonacci polynomials. The Gauss Lucas polynomials are described and the relation with Lucas polynomials are explained. We show that there is a relation between the Gauss Fibonacci polynomials and the Fibonacci polynomials. The Gauss Lucas polynomials are also given by using the Gauss Fibonacci polynomials. Some theorems like Cassini's theorem are proved for the polynomials. Their Binet's formulas are obtained. We also define the matrices of the Gauss Fibonacci polynomials and the Gauss Lucas polynomials. We examine properties of the matrices.Catalan transform of the incomplete Jacobsthal numbers and incomplete generalized Jacobsthal polynomialshttps://zbmath.org/1491.110232022-09-13T20:28:31.338867Z"Özkan, Engin"https://zbmath.org/authors/?q=ai:ozkan.engin"Uysal, Mine"https://zbmath.org/authors/?q=ai:uysal.mine"Kuloğlu, Bahar"https://zbmath.org/authors/?q=ai:kuloglu.baharSums of finite products of Pell polynomials in terms of hypergeometric functionshttps://zbmath.org/1491.110242022-09-13T20:28:31.338867Z"Patra, Asim"https://zbmath.org/authors/?q=ai:patra.asim"Panda, Gopal Krishna"https://zbmath.org/authors/?q=ai:panda.gopal-krishnaThe purpose of this article is to express sums of finite products of Pell polynomials \(P_n(x)\) in terms of hypergeometric functions. First, a fundamental connection between the Chebyshev polynomials of the second kind and the Pell polynomials is established. The new Chebyshev polynomials of the third and fourth kind as well as other special polynomials expressed in terms of hypergeometric functions are used in the proofs. Several special integrals and generating functions are used in the proofs. As special case, a formula for the \(r\)-th derivative of the Pell polynomial is derived. Obviously, many of these polynomials can be expressed in terms of each other.
Reviewer: Thomas Ernst (Uppsala)The generalized Lucas hybrinomials with two variableshttps://zbmath.org/1491.110252022-09-13T20:28:31.338867Z"Sevgi, Emre"https://zbmath.org/authors/?q=ai:sevgi.emreSummary: Özdemir defined the hybrid numbers as a generalization of complex, hyperbolic and dual numbers. In this research, we define the generalized Lucas hybrinomials with two variables. Also, we get the Binet formula, generating function and some properties for the generalized Lucas hybrinomials. Moreover, Catalan's, Cassini's and d'Ocagne's identities for these hybrinomials are obtained. Lastly, by the help of the matrix theory we derive the matrix representation of the generalized Lucas hybrinomials.Split complex bi-periodic Fibonacci and Lucas numbershttps://zbmath.org/1491.110262022-09-13T20:28:31.338867Z"Yilmaz, Nazmiye"https://zbmath.org/authors/?q=ai:yilmaz.nazmiyeSummary: The initial idea of this paper is to investigate the split complex bi-periodic Fibonacci and Lucas numbers by using SCFLN now on. We try to show some properties of SCFLN by taking into account the properties of the split complex numbers. Then, we present interesting relationships between SCFLN.Partial factorizations of products of binomial coefficientshttps://zbmath.org/1491.110272022-09-13T20:28:31.338867Z"Du, Lara"https://zbmath.org/authors/?q=ai:du.lara"Lagarias, Jeffrey C."https://zbmath.org/authors/?q=ai:lagarias.jeffrey-cLet \[\overline{G}_n =\prod_{k=0}^n \binom{n}{k},\] the product of the elements of the \(n\)th row of Pascal's triangle.
The authors study the partial factorizations of \(\overline{G}_n\) given by the product \(G(n, x)\) of all prime factors \(p\) of \(\overline{G}_n\) having \(p\leq x\), counted with multiplicity.
They show \[\log G(n, \alpha n)f_G(\alpha)n^2\quad\text{as } n \to \infty\] for a limit function \(f_G(\alpha)\) defined for \(0\leq\alpha\leq 1\). The the main results are deduced from study of functions \(A(n, x)\), \(B(n, x)\), that encode statistics of the base \(p\) radix expansions of the integer \(n\) (and smaller integers), where the base \(p\) ranges over primes \(p\leq x\). Asymptotics of \(A(n, x)\) and \(B(n, x)\) are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.
Reviewer: Guo-Shuai Mao (Nanjing)On Motzkin numbers and central trinomial coefficientshttps://zbmath.org/1491.110282022-09-13T20:28:31.338867Z"Sun, Zhi-Wei"https://zbmath.org/authors/?q=ai:sun.zhiwei|sun.zhi-wei.1|sun.zhi-weiSummary: The Motzkin numbers \(M_n = \sum^n_{k = 0} \binom{n}{2 k} \binom{2 k}{k} / (k + 1) (n = 0, 1, 2, \ldots )\) and the central trinomial coefficients \(T_n (n = 0, 1, 2, \ldots )\) given by the constant term of \(( 1 + x + x^{- 1} )^n\), have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with \(n\) any positive integer:
\[
\begin{aligned}
\frac{ 2}{ n} &\sum_{k = 1}^n(2 k + 1) M_k^2 \in \mathbb{Z},\\
\frac{ n^2 ( n^2 - 1 )}{ 6} &\Bigg| \sum_{k = 0}^{n - 1} k(k + 1)(8 k + 9) T_k T_{k + 1},
\end{aligned}
\]
and also
\[
\sum_{k = 0}^{n - 1}(k + 1)(k + 2)(2 k + 3) M_k^2 3^{n - 1 - k} = n(n + 1)(n + 2) M_n M_{n - 1}.
\]Asymptotic behavior of Bernoulli-Dunkl and Euler-Dunkl polynomials and their zeroshttps://zbmath.org/1491.110292022-09-13T20:28:31.338867Z"Mínguez Ceniceros, Judit"https://zbmath.org/authors/?q=ai:minguez-ceniceros.judit"Varona, Juan L."https://zbmath.org/authors/?q=ai:varona-malumbres.juan-luisThe purpose of this article is to study the asymptotic behavior of the recently introduced Bernoulli-Dunkl and Euler-Dunkl polynomials.
These polynomials are defined with so-called \(E_{\alpha}\) functions instead of exponential functions in the numerator. Dilcher showed asympotic behaviour of the Bernoulli and the Euler polynomials in terms of trigonometric functions and the authors show corresponding asymptotic behavior of the Bernoulli-Dunkl and the Euler-Dunkl polynomials. The coefficients of the asymptotic expansions as well as the right hand sides are Bessel functions. The first positive zero of \(J_{\alpha}(z)\) appears as coefficients in function arguments as well as in numerators and denominators. Several formulas for Bernoulli-Dunkl polynomials, which are similar to Bernoulli polynomials, where the Dunkl operator replaces the derivative, and new coefficients replace the binomial coefficients are used to prove the main theorems. Graphs are shown which display the differences between the new polynomials and Bernoulli polynomials. And similar for Euler-Dunkl polynomials. The zeros for Bernoulli-Dunkl and Euler-Dunkl polynomials are shown in graphs.
Reviewer: Thomas Ernst (Uppsala)Semiorthogonality of geometric polynomialshttps://zbmath.org/1491.110302022-09-13T20:28:31.338867Z"Kargın, Levent"https://zbmath.org/authors/?q=ai:kargin.levent"Çay, Emre"https://zbmath.org/authors/?q=ai:cay.emreGeometric polynomials are also termed as Fubini polynomials which is most commonly used in literature.
Special polynomials and numbers have significant roles in various branches of mathematics, theoretical physics, and engineering. The problems arising in mathematical physics and engineering are framed in terms of differential equations. Most of these equations can only be treated by using various families of special polynomials which provide new means of mathematical analysis. They are widely used in computational models of scientific and engineering problems. In addition, these special polynomials allow the derivation of different useful identities in a straightforward way and help in introducing new families of special polynomials.
The Fubini-type polynomials (or geometric-type polynomials) appear in combinatorial mathematics and play an important role in the theory and applications of mathematics, thus many number theory and combinatorics experts have extensively studied their properties and obtained series of interesting results. The Fubini-type numbers and polynomials are related Bernoulli numbers with diverse extensions and proven to be an effective tool in different topics in combinatorics and analysis.
In the related paper, the authors have examined the semiorthogonality of Fubini and higher order Fubini polynomials. They showed that the integrals of products of the higher order Fubini polynomials can be evaluated in terms of Bernoulli numbers, which means that the higher order geometric polynomials are also semiorthogonal. As applications, they have given some new explicit formulas for Bernoulli and \(p\)-Bernoulli numbers.
Reviewer: Uğur Duran (Iskenderun)2-variable Fubini-degenerate Apostol-type polynomialshttps://zbmath.org/1491.110312022-09-13T20:28:31.338867Z"Nahid, Tabinda"https://zbmath.org/authors/?q=ai:nahid.tabinda"Ryoo, Cheon Seoung"https://zbmath.org/authors/?q=ai:ryoo.cheon-seoungSpectral theory of regular sequenceshttps://zbmath.org/1491.110322022-09-13T20:28:31.338867Z"Coons, Michael"https://zbmath.org/authors/?q=ai:coons.michael"Evans, James"https://zbmath.org/authors/?q=ai:evans.james-a|evans.james-b|evans.james-e|evans.james-w|evans.james-r|evans.james-d"Mañibo, Neil"https://zbmath.org/authors/?q=ai:manibo.neilRegular sequences are a well-studied generalization to arbitrary (unbounded) alphabets of fixed points of constant-length substitutions (also called morphisms) defined over a finite alphabet, that is, of automatic sequences. They can be defined by the fact that the vector space generated by the \(k\)-kernel of the sequence is finite-dimensional.
From the abstract: ``Using the harmonic analysis of measures associated with substitutions as motivation, we study the limiting asymptotics of regular sequences by constructing a systematic measure-theoretic framework surrounding them. The constructed measures are generalisations of mass distributions supported on attractors of iterated function systems.''
Reviewer: Michel Rigo (Liège)On an elliptic curve involving pairs of triangles and special quadrilateralshttps://zbmath.org/1491.110332022-09-13T20:28:31.338867Z"Li, Yangcheng"https://zbmath.org/authors/?q=ai:li.yangcheng"Zhang, Yong"https://zbmath.org/authors/?q=ai:zhang.yong.4Some exponential Diophantine equations. II: The equation \(x^2 - dy^2=k^z\) for even \(k\)https://zbmath.org/1491.110342022-09-13T20:28:31.338867Z"Fujita, Yasutsugu"https://zbmath.org/authors/?q=ai:fujita.yasutsugu-fujita"Le, Maohua"https://zbmath.org/authors/?q=ai:le.maohuaAuthors' abstract: Let \(D\) be a nonsquare integer, and let \(k\) be an integer with \(|k|\geq 1\) and \(\gcd(D,k)=1\). In the part \(I\) of this paper, using some properties on the representation of integers by binary quadratic primitive forms with discriminant \(4D\), the second author [Part I, J. Number Theory 55, No. 2, 209--221 (1995; Zbl 0852.11015)] gave a series of explicit formulas for the integer solutions \((x, y, z)\) of the equation \(x^2-Dy^2=k^z\), \(\gcd(x,y)=1\), \(z>0\) for \(2\nmid k\) or \(|k|\) is a power of 2. In this part, we give similar results for the other cases of \(k\).On the Diophantine equation \(x^2+b^m=c^n\) with \(a^2+b^4=c^2\)https://zbmath.org/1491.110352022-09-13T20:28:31.338867Z"Terai, Nobuhiro"https://zbmath.org/authors/?q=ai:terai.nobuhiroThe paper under review concerns a problem about exponential Diophantine equations. More precisely, let \(a, b\) and \(c\) be pairwise relatively prime positive integers such that \(a^2+b^4=c^2\) and \(b\) is odd. Under some conditions, the main result of the paper is that the Diophantine equation \(x^2+b^m=c^n\) has only one positive integer solution triple \((x, m, n)=(a, 4, 2)\).
One of the interesting problems in number theory which remains open is the Jeśmanowicz' Conjecture. Variations of this conjecture are widely studied. The motivation of the paper is coming from an analogue of Jeśmanowicz' Conjecture.
The author proposed another similar conjecture and he verified the conjecture in a fixed range with the Magma Computational Algebra System.
The proof is split into cases and based on calculations and a series of lemmas.
Reviewer: Ilker Inam (Bilecik)An exponential Diophantine equation related to odd perfect numbershttps://zbmath.org/1491.110362022-09-13T20:28:31.338867Z"Yamada, Tomohiro"https://zbmath.org/authors/?q=ai:yamada.tomohiroConsider the Diophantine equation \[\dfrac{x^\ell-1}{x-1}=p^m q,\quad m\ge 0.\tag{1.1} \] In this work, the author gives a stronger upper bound for the number of solutions of (1.1). He proved that if \(p,q,\ell\) are fixed primes such that \(\ell\ge 17\) and \(p\equiv q \equiv 1 \pmod{\ell}\), then (1.1) has at most four positive integral solutions \((x,m)\). Furthermore, combining this result with an argument in [\textit{T. Yamada}, Colloq. Math. 156, No. 1, 15--23 (2019; Zbl 1457.11011); Colloq. Math. 162, No. 2, 311--313 (2020; Zbl 1484.11008)], he gives an application to the odd perfect number problem.
In the proofs, he uses Baker's theory which is based on a lower bound for linear forms of logarithms due to \textit{E. M. Matveev} [Izv. Math. 64, No. 6, 1217--1269 (2000; Zbl 1013.11043); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 64, No. 6, 125--180 (2000)] and some lemmas in [\textit{T. Yamada}, Colloq. Math. 156, No. 1, 15--23 (2019; Zbl 1457.11011); Colloq. Math. 162, No. 2, 311--313 (2020; Zbl 1484.11008)].
Reviewer: Gökhan Soydan (Bursa)Weighted sums of generalized polygonal numbers with coefficients \(1\) or \(2\)https://zbmath.org/1491.110372022-09-13T20:28:31.338867Z"Kim, Daejun"https://zbmath.org/authors/?q=ai:kim.daejunDefine
\[
P_{m,a}(x):= \sum^k_{i=1} a_i P_m(x_i)
\]
for \(m\)-goal numbers \(P_m(x)\). The main result is
Theorem 1.1. For any positive integer \(m\ge 10\), the sum \(P_{m,(1^\alpha,2^\beta)}\) is universal if and only if it represents \(1\), \(m-4\), and \(m-2\). Moreover, the sum \(P_{m,(1^\alpha,2^\beta)}\) is universal if and only if it represents \(1,3,5,10,19\), and \(23\) if \(m=7\), and \(1,5,7\), and \(34\) if \(m=9\).
Reviewer: Meinhard Peters (Münster)The number of representations of integers by 4-dimensional strongly \(N\)-modular latticeshttps://zbmath.org/1491.110382022-09-13T20:28:31.338867Z"Jung, Ho Yun"https://zbmath.org/authors/?q=ai:jung.ho-yun"Kim, Chang Heon"https://zbmath.org/authors/?q=ai:kim.chang-heon"Kim, Kyoungmin"https://zbmath.org/authors/?q=ai:kim.kyoungmin"Kwon, Soonhak"https://zbmath.org/authors/?q=ai:kwon.soonhakIn this work, the authors proved that a space of cusp forms of weight \(2\) with level \(N\)and real character \(\chi \) has dimension \(1\) if and only if \(\chi \) is trivial and
\[
N\in\{11,14,15,17,19,20,21,24,27,32,36,49\}
\]
and derive bases for spaces of cusp forms of weight \(2\) with trivial character and
\[
N\in \{17,19,21,49\}.
\]
Further they provided formulas for the number of representations of integers by \(4\)-dimensional strongly \(N\)-modular lattices for
\[
N\in \{11,14,15,17,19,20,21,24,27,32,36\}.
\]
Reviewer: Ahmet Tekcan (Bursa)A cubic ring of integers with the smallest Pythagoras numberhttps://zbmath.org/1491.110392022-09-13T20:28:31.338867Z"Krásenský, Jakub"https://zbmath.org/authors/?q=ai:krasensky.jakubThe author proves that the ring of integers in the unique number field \(K^{(49)}= \mathbb{Q}(\zeta_7+ \zeta^{-1}_7)\) of discriminant \(49\) has Pythagoras number equal to \(4\). Moreover, it is determined which numbers are sums of integral squares in this field. An application is presented in the construction of a diagonal universal quadratic form in \(5\) variables, namely \(x^2_1+ x^2_2+ x^2_3+ x^2_4+ (1+\rho+\rho^2)x^2_5\), where \(\varphi= \zeta_7+ \zeta^{-1}_7\).
Reviewer: Meinhard Peters (Münster)On the representation of integers by binary forms defined by means of the relation \((x + yi)^n= R_n(x,y) + J_n(x,y)i\)https://zbmath.org/1491.110402022-09-13T20:28:31.338867Z"Mosunov, Anton"https://zbmath.org/authors/?q=ai:mosunov.antonLet \(F\) be a binary form with integer coefficients of degree \(d \ge 3\) and nonzero discriminant. Let \(R_F(Z)\) denote the number of integers of absolute value at most \(Z\) which are represented by \(F\).\par \textit{C. L. Stewart} and \textit{S. Y. Xiao} [Math. Ann. 375, No. 1--2, 133--163 (2019; Zbl 1464.11035)] proved that \(R_F(Z)\sim C_F Z^{2/d}\) for some positive number \(C_F\). For a number of families of binary forms with integer coefficients the value of \(C_F\) is either known or well estimated.\par Let \(R_n(x,y)\) and \(J_n(x,y)\) be defined by \[(x+iy)^n=R_n(x,y)+J_n(x,y)i.\] The author discusses properties of \(R_n(x,y)\) and \(J_n(x,y)\) and determines \(C_{R_n}\) and \(C_{J_n}\) for these binary forms.
Reviewer: István Gaál (Debrecen)Uniform bounds in Waring's problem over some diagonal formshttps://zbmath.org/1491.110412022-09-13T20:28:31.338867Z"Pliego, Javier"https://zbmath.org/authors/?q=ai:pliego.javierThis technical paper extends classical Waring's problem to representation of large natural numbers \(n\) as \[ n = \sum_{j=1}^s x_j^\ell, \] where the \(x_j\) belong to some special sparse set. The circle method is used for the proofs.
Reviewer: Luis Gallardo (Brest)A proof of the mod 4 unimodal sequence conjectures and related mock theta functionshttps://zbmath.org/1491.110422022-09-13T20:28:31.338867Z"Chen, Rong"https://zbmath.org/authors/?q=ai:chen.rong"Garvan, F. G."https://zbmath.org/authors/?q=ai:garvan.frank-gThe authors provide detailed background and required basic necessary results needed for this article. A mod \(4\) unimodal sequence conjecture established by \textit{J.Bryson} [``Unimodal sequences and quantum and mock modular forms'', Proc. Natl. Acad. Sci. USA 109, No. 40, 16063--16067 (2012; \url{doi:10.1073/pnas.1211964109})] is proved by the authors. Another mod \(4\) unimodal sequence conjecture given by \textit{B. Kim} et al. [Proc. Am. Math. Soc. 144, No. 9, 3687--3700 (2016; Zbl 1404.11052)] is pointed out as \textit{false}, and suggested its correct form and proof as Theorem 4.1. Authors also established an analogous mod \(4\) results which state as spt-function and for the coefficients of related mock theta function, introduced by \textit{G. E. Andrews} [J. Reine Angew. Math. 624, 133--142 (2008; Zbl 1153.11053)]. Several related new results are also discussed in this paper, which are useful for further advancement of the subject.
Reviewer: M. P. Chaudhary (New Delhi)On the zeros of period functions associated to the Eisenstein series for \(\Gamma_0^+(N)\)https://zbmath.org/1491.110432022-09-13T20:28:31.338867Z"Choi, SoYoung"https://zbmath.org/authors/?q=ai:choi.soyoung"Im, Bo-Hae"https://zbmath.org/authors/?q=ai:im.bo-haeFor a positive integer \(N\), let \(\Gamma_0(N)^+\) be the modular group generated by the elements of \(\Gamma_0(N)\) and the Fricke involution \(W_N\). Let \(E_{k}^+(z)\) be the Eisenstein series of even weight \(k\) for \(\Gamma_0(N)^+\). For a modular form \(f(z)=\sum_{n=0}^\infty a(n)e^{2\pi inz}\) of weight \(k\), the Eichler integral of \(f\) is defined by \(\mathcal{E}_f(z)=\sum_{n=0}^\infty a(n)n^{1-k}e^{2\pi inz}\). As an analogy of the period function of Eisenstein series for \(\mathrm{SL}_2(\mathbb Z)\), the authors define the period function of \(E_{k}^+\) by \(r(E_{k}^+;z)=-\frac{\Gamma(k-1)}{(2\pi i)^{k-1}}(\mathcal{E}_{E_{k}^+}-\mathcal{E}_{E_{k}^+}|_{2-k}W_N(z))\). The function \(r(E_{k}^+;z)-((k-1)N^{k/2}z)^{-1}\) is a polynomial of \(z\) of degree \(k-1\). From \(r(E_{k}^+;z)\), they consider three polynomials \(R_k(z)=(k-1)zr(E_{k}^+;z), R_k^-(z)=(k-1)zr^-(E_{k}^+;z)\) and \(P_k(z)\), where \(r^-(E_{k}^+;z)\) is the odd part of \(r(E_{k}^+;z)\) and, up to a nonzero constant, \(P_k(z)\) is defined by \(r(E_{k}^+;z)-(z^{k-1}+\frac{1}{N^{k/2}z})/(k-1)\). The polynomials \(R_k^-\) and \(R_k\) are of degree \(k\) and \(P_k\) is of degree \(k-2\).
The authors are interested in the location of zeros of these three polynomials and show that all zeros of \(P_k\) (resp. \(R_k^-\) and \(R_k\)) lie on the circle \(|z|=1/\sqrt{N}\), if \(N\ge 3, k\ge 4\) (resp. \(N\ge 5, k\ge 10 \) and \(N\ge 17, k\ge 10\)), and the zeros of each polynomial are simple and regularly distributed on this circle. Further, if \(N=1,k\ge 4\) and \(N=2, k\ge 1\), then all zeros of \(P_k\) lie on the circle \(|z|=1/\sqrt{N}\). If \(N=2,3,4\) and \(k\ge 4\), then \(R_k^-\) has exactly \(k-4\) zeros lying on the circle \(|z|=1/\sqrt N\) and four distinct real zeros lying outside the circle. The key point of the proof is the \(N\)-self-inversive property of \(P_k,R_k\) and \(R_k^-\). Here a polynomial \(P(z)\) of degree \(d\) is \(N\)-self-inversive if \(P\) satisfies \( P(z)=\epsilon (\sqrt Nz)^dP(1/Nz)\) for some \(\epsilon\in\mathbb C\). For \(N\)-self-inversive polynomial \(P(z)=\sum_{j=0}^dA_jz^j\in\mathbb C[z]\), if \(|A_d|\ge \frac12 \sum_{j=1}^{d-1}|A_j|\sqrt N^{d-1}\), then all of zeros of \(P\) lie on the circle \(|z|=1/\sqrt N\). Therefore, they check the above inequality for the coefficients of the polynomials under consideration. Further to show that the zeros are simple and regularly distributed on the circle, they deal with many fine inequalities.
Reviewer: Noburo Ishii (Kyoto)The Eichler integral of \(E_2\) and \(q\)-brackets of \(t\)-hook functionshttps://zbmath.org/1491.110442022-09-13T20:28:31.338867Z"Ono, Ken"https://zbmath.org/authors/?q=ai:ono.kenSummary: For functions \(f: \mathcal{P}\mapsto \mathbb{C}\) on partitions, \textit{S. Bloch} and \textit{A. Okounkov} [Adv. Math. 149, No. 1, 1--60 (2000; Zbl 0978.17016)] defined a power series \(\langle f \rangle_q\) that is the ``weighted average'' of \(f\). As Fourier series in \(q = e^{2 \pi iz}\), such \(q\)-brackets generate the ring of quasimodular forms, and the modular forms that are powers of Dedekind's eta-function. Using work of Berndt and Han, we build modular objects from
\[
f_t(\lambda):= t\sum_{h\in \mathcal{H}_t(\lambda)}\frac{1}{h^2},
\]
weighted sums over partition hook numbers that are multiples of \(t\). We find that \(\langle f_t \rangle_q\) is the Eichler integral of \((1 - E_2(tz))/24\), which we modify to construct a function \(M_t(z)\) that enjoys weight 0 modularity properties. As a consequence, the non-modular Fourier series
\[
H_t^\ast(z):=\sum_{\lambda \in \mathcal{P}} f_t(\lambda)q^{|\lambda |-\frac{1}{24}}
\]
inherits weight \(- 1/2\) modularity properties. These are sufficient to imply a Chowla-Selberg-type result, generalizing the fact that weight \(k\) algebraic modular forms evaluated at discriminant \(D < 0\) points \(\tau\) are algebraic multiples of \(\Omega_D^k\), the \(k\)th power of the canonical period. If we let \(\Psi (\tau):=-\pi i \left (\frac{\tau^2-3\tau +1}{12\tau}\right)-\frac{\log (\tau)}{2}\), then for \(t = 1\) we prove that
\[
\begin{aligned} H_1^\ast(-1/\tau)-\frac{1}{\sqrt{-i\tau}}\cdot H_1^\ast(\tau)\in \overline{\mathbb{Q}}\cdot \frac{\Psi (\tau)}{\sqrt{\Omega_D}}. \end{aligned}
\]
For the entire collection see [Zbl 1479.47003].Non-virtually abelian anisotropic linear groups are not boundedly generatedhttps://zbmath.org/1491.110452022-09-13T20:28:31.338867Z"Corvaja, Pietro"https://zbmath.org/authors/?q=ai:corvaja.pietro"Rapinchuk, Andrei S."https://zbmath.org/authors/?q=ai:rapinchuk.andrei-s"Ren, Jinbo"https://zbmath.org/authors/?q=ai:ren.jinbo"Zannier, Umberto M."https://zbmath.org/authors/?q=ai:zannier.umberto-mThis is an important paper which introduces new methods for dealing with problems involving the bounded generation of groups. A group \(G\) is said to be \textit{boundedly generated} or having property (BG) if there exists a finite subset \(X\) of \(G\) and a positive integer \(m\) such that each element \(g\) of \(G\) can written in the form \[g=x_1^{n_1}\cdots x_m^{n_m},\] where \(x_i \in X\) and \( n_i \in \mathbb{Z}\). By definition every group having (BG) is finitely generated. The simplest examples are finitely generated virtually nilpotent groups. The finitely generated groups which do not have (BG) include all non-cyclic free groups of finite rank. The first examples of \textit{non-virtually solvable} groups having (BG) are due to \textit{D. Carter} and \textit{G. Keller} [Am. J. Math. 105, 673--687 (1983; Zbl 0525.20029)]. They show that \(\mathrm{SL}_n(\mathcal{O})\), where \(\mathcal{O}\) is a ring of algebraic integers and \(n \geq 3\), is boundedly generated and the bounding number \(m\) is determined by \(n\) and the discriminant of \(\mathcal{O}\). In addition they show that \(X\) can be chosen to consist entirely of elementary (and hence unipotent) matrices. This result (which does not hold in general for \(n=2\)) has subsequently been extended to all Chevalley groups of rank \(>1\) and most quasi-split groups. Although bounded generation is a purely combinatorial property of groups it has surprisingly many important consequences and applications. For example it is known that bounded generation for \(S\)-arithmetic subgroups of absolutely almost simple algebraic groups over a number field \(K\), where \(S\) is a finite set of valuations of \(K\) containing all the non-Archimedean valuations, ensures that such groups have the \textit{congruence subgroup property} (subject to some natural assumptions). More precisely this means that for these groups the \textit{congruence kernel,} as originally defined by Serre, is \textit{finite}.
The property (BG) plays a crucial role in the proof of the Margulis-Zimmer conjecture for commensurated subgroups of higher rank \(S\)-arithmetic subgroups of Chevalley groups as well as the estimation of Kazhdan constants. In addition the natural extension of property (BG) to profinite groups, \((\mathrm{BG})_{\mathrm{pr}}\), plays a significant role. In particular the pro-\(p\) groups having \((\mathrm{BG})_{\mathrm{pr}}\) are precisely the \(p\)-adic analytic groups.
A linear group \(\Gamma \subset \mathrm{GL}(K)\), where \(K\) is a field of characteristic zero, is said to be \textit{anisotropic} if \(\Gamma\) consists entirely of semi-simple elements. To date all the \(S\)-arithmetic subgroups of absolutely almost simple algebraic groups over a number field \(K\) known to be boundedly generated are not anisotropic. One of the principal aims of this paper is to find examples of boundedly generated linear groups which are generated by semi-simple elements. The main results follow from the following.
Theorem A. Let \(\Gamma \subset \mathrm{GL}_n(K)\) be a linear group over a field \(K\) of characteristic zero which is not virtually solvable. Then in every possible presentation (BG) for \(\Gamma\) at least two of the elements \(x_i\) are not semi-simple.
After reducing to the case where \(K\) is a number field the proof follows from a technical result involving a finite set of matrices in \(\mathrm{GL}_n(\overline{\mathbb{Q}})\) whose eigenvalues satisfy a condition called \textit{multiplicatively independence}. The proof which is long and intricate makes use of properties of \textit{generic elements} in Zariski-dense subgroups as well as Laurent's theorem from Diophantine geometry. This has a number of important consequences.
Corollary B. An anisotropic linear group \(\Gamma \subset \mathrm{GL}_n(K)\) over a field of characteristic zero has (BG) if and only if it is finitely generated and virtually abelian.
Theorem C. Let \(G\) be an algebraic group over a number field \(K\) and let \(S\) be a finite set of valuations of \(K\) containing all the archimedean ones. Then the infinite \(S\)-arithmetic subgroups of \(G\), where \(G\) is absolutely almost simple and \(K\)-anisotropic, are not boundedly generated.
It is clear that by definition if \(\Gamma\) is a discrete group having property (BG) the its profinite completion \(\widehat{\Gamma}\) has property \((\mathrm{BG})_{\mathrm{pr}}\). The converse does not hold.
Corollary D. There exist residually finite finitely generated groups \(\Gamma\) not having property \(\mathrm{(BG)}\) for which \(\widehat{\Gamma}\) has property \((\mathrm{BG})_{\mathrm{pr}}\).
Diophantine geometry together with the existence of generic elements can provide an apparently novel way to determine whether (or not) a group is boundedly generated. In this paper this approach has extended existing results. For example a standard method for proving a group does \textit{not} have (BG) reduces to showing that its second bounded cohomology is an infinite dimensional real vector space. Theorem C proves the same result for some groups whose second bounded cohomology vanishes. It is hoped, for example, that these methods can provide a verifiable sufficient condition for some free amalgamated products to have bounded generation.
Reviewer: Alexander W. Mason (Glasgow)The maximal discrete extension of the Hermitian modular grouphttps://zbmath.org/1491.110462022-09-13T20:28:31.338867Z"Krieg, Aloys"https://zbmath.org/authors/?q=ai:krieg.aloys"Raum, Martin"https://zbmath.org/authors/?q=ai:raum.martin"Wernz, Annalena"https://zbmath.org/authors/?q=ai:wernz.annalenaLet \(\mathbb{K}\) be an\ imaginary-quadratic number field and \(\Gamma _{n}\left( \mathcal{O}_{\mathbb{K}}\right) \) be the Hermitian modular group of degree \(n\) over \(\mathbb{K}\). The main purpose of the paper under review is to determine the maximal discrete extension of \(\Gamma _{n}\left(\mathcal{O}_{\mathbb{K}}\right) \) in SU\(\left( n,n;\mathbb{C}\right) \). The case \(n=2\) is examined in detail. The extended Hermitian modular group of degree \(2\) is described explicitly by means of generalized Atkin-Lehner involutions and a natural characterization of this group in SO\((2,4)\) is found.
Reviewer: Nihal Yılmaz Özgür (Balikesir)Cubic level analogue of Ramanujan's Eisenstein series identitieshttps://zbmath.org/1491.110472022-09-13T20:28:31.338867Z"K. R., Vasuki"https://zbmath.org/authors/?q=ai:k-r.vasuki"A., Darshan"https://zbmath.org/authors/?q=ai:a.darshanSummary: Let \(Q_n=1+240\sum_{k=1}^{\infty}\frac{k^3q^{nk}}{1-q^{nk}}\). On page 51--53 of his lost notebook, Ramanujan recorded very interesting identities which relates \(Q_1,Q_5,Q_7\) with his theta functions. In this article, we establish analogous identities with respect to \(Q_1\) and \(Q_3\).Three-parameter mock theta functionshttps://zbmath.org/1491.110482022-09-13T20:28:31.338867Z"Cui, Su-Ping"https://zbmath.org/authors/?q=ai:cui.su-ping"Gu, Nancy S. S."https://zbmath.org/authors/?q=ai:gu.nancy-shan-shan"Hou, Qing-Hu"https://zbmath.org/authors/?q=ai:hou.qinghu"Su, Chen-Yang"https://zbmath.org/authors/?q=ai:su.chenyangSummary: Mock theta functions were first introduced by Ramanujan. Historically, mock theta functions can be represented as Eulerian forms, Appell-Lerch sums, Hecke-type double sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, in view of the \(q\)-Zeilberger algorithm and the Watson-Whipple transformation formula, we establish five three-parameter mock theta functions in Eulerian forms, and express them by Appell-Lerch sums. Especially, the main results generalize some two-parameter mock theta functions. For example, setting \((m, q, x) \to(1, q^{1 / 2}, x q^{- 1 / 2})\) in
\[
\sum_{n = 0}^\infty \frac{ (- q^2; q^2)_n q^{n^2 + (2 m - 1) n}}{ (x q^m, x^{- 1} q^m; q^2)_{n + 1}},
\] we derive the universal mock theta function \(g_2(x, q)\).Modular iterated integrals associated with cusp formshttps://zbmath.org/1491.110492022-09-13T20:28:31.338867Z"Diamantis, Nikolaos"https://zbmath.org/authors/?q=ai:diamantis.nikolaosThe paper under review brings two topics together, namely, modular iterated integrals and higher-order modular forms and it looks very interesting. The main contribution of the paper is providing an explicit family of modular iterated integrals which involves cusp forms. This construction is based on an extension of higher-order modular forms.
Reviewer: Ilker Inam (Bilecik)Module constructions for certain subgroups of the largest Mathieu grouphttps://zbmath.org/1491.110502022-09-13T20:28:31.338867Z"Beneish, Lea"https://zbmath.org/authors/?q=ai:beneish.leaSummary: For certain subgroups of \(M_{24}\), we give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms. These meromorphic Jacobi forms are canonically associated to the mock modular forms of Mathieu moonshine. The construction is related to the Conway moonshine module and employs a technique introduced by Anagiannis-Cheng-Harrison [\textit{V. Anagiannis} et al., Commun. Math. Phys. 366, No. 2, 647--680 (2019; Zbl 1411.81157)]. With this construction we are able to give concrete vertex algebraic realizations of certain cuspidal Hecke eigenforms of weight two. In particular, we give explicit realizations of trace functions whose integralities are equivalent to divisibility conditions on the number of \(\mathbb{F}_p\) points on the Jacobians of modular curves.Quantum Jacobi forms and sums of tails identitieshttps://zbmath.org/1491.110512022-09-13T20:28:31.338867Z"Folsom, Amanda"https://zbmath.org/authors/?q=ai:folsom.amanda-l"Pratt, Elizabeth"https://zbmath.org/authors/?q=ai:pratt.elizabeth"Solomon, Noah"https://zbmath.org/authors/?q=ai:solomon.noah"Tawfeek, Andrew R."https://zbmath.org/authors/?q=ai:tawfeek.andrew-rSo-called ``sums of tails'' identities have a long history, going back to Ramanujan, but have seen much recent interest in the modular forms literature due to their connections to mock modular and quantum modular forms. A distinguished sum of tails identity is:
\[
\sum_{n\geq0}((q;q)_{\infty}-(q;q)_n)=(q;q)_{\infty}\left(-\frac12+\sum_{n\geq1}\frac{q^n}{1-q^n} \right)+\frac12\sum_{n\geq1}\left(\frac{12}{n}\right)nq^{\frac{n^2-1}{24}},
\]
which Zagier used in his pathbreaking study of the Kontsevich function \(\sum_{n\geq0}(q;q)_n\). This allowed Zagier to study its transformation properties, giving an early example of a quantum modular form and in particular to study asymptotics of Vassiliev invariants in knot theory.
These quantum modular forms have features similar to modular forms, but instead of being defined on the upper half plane, they are defined on the cusps (namely, on \(\mathbb Q\cup\{i\infty\}\)). To allow for interesting examples, they do not transform strictly as modular forms, but their errors to modularity (differences between two sides in the usual modular symmetry relation) are in some sense ``nice''. This includes properties such as becoming defined on all of \(\mathbb R\), and being smooth, \(\mathcal C^k\), real-analytic, etc.
Quantum modular forms were codified in Zagier's seminal 2010 Clay Math Proceedings paper
[\textit{D. Zagier}, Clay Math. Proc. 11, 659--675 (2010; Zbl 1294.11084)]. In the years since, this has grown into a large subject with applications throughout math and physics, and many further examples have been uncovered.
In [Arch. Math. 107, No. 4, 367--378 (2016; Zbl 1347.05012)], \textit{K. Bringmann} and the first author defined quantum Jacobi forms, which can encode an infinite family of quantum modular forms into a single two-variable function. This has also developed into a topic with many applications and examples in the literature.
The current paper is a tour de force of new identities and examples. Firstly, two variable versions of sums of tails identities are established, which are used to produce quantum Jacobi forms in an analogous way to Zagier's original construction in the one-variable quantum modular case. The authors further establish interesting radial asymptotic identities and provide formulas for such expansions in terms of \(L\)-values.
This provides a plethora of new information which recovers many previous examples in the literature, and is a valuable resource for any reader looking to learn more about quantum modular forms or to find useful tools or examples for applications of this subject.
Reviewer: Larry Rolen (Dublin)Nonvanishing of kernel functions and Poincaré series for Jacobi formshttps://zbmath.org/1491.110522022-09-13T20:28:31.338867Z"Pandey, Shivansh"https://zbmath.org/authors/?q=ai:pandey.shivansh"Sahu, Brundaban"https://zbmath.org/authors/?q=ai:sahu.brundabanSummary: Y. Martin introduced a set of kernel functions for the Jacobi group to study \(2m\) Dirichlet series associated with a Jacobi form of weight \(k\) and index \(m\). We study nonvanishing of these kernel functions, Poincaré series and also study nonvanishing of \(2m\) Dirichlet series associated with Jacobi form of weight \(k\) and index \(m\).Langlands reciprocity: \(L\)-functions, automorphic forms, and diophantine equationshttps://zbmath.org/1491.110532022-09-13T20:28:31.338867Z"Emerton, Matthew"https://zbmath.org/authors/?q=ai:emerton.matthewThe article under review, a chapter in [\textit{J. Mueller} (ed.) and \textit{F. Shahidi} (ed.), The genesis of the Langlands program. Cambridge: Cambridge University Press (2021; Zbl 1472.11005)], gives a description of the theory of reciprocity laws in algebraic number theory and its relation to the theory of \( L \)-functions.
The author starts with a brief (quasi-)historical introduction, from the genesis of the word \textit{reciprocity} itself, going to the early history of \( \zeta \)- and \( L \)-functions -- the pioneering works of Euler, Riemann and Dirichlet -- the subsequent introduction by Dedekind and Hecke of their respective \( L \)-functions, and their recast into the modern adélique language of Chevalley by Tate in his thesis, which not only reproved the results of Hecke, but unified the arguments for proving 1) the existence of Euler products for Hecke \( L \)-functions 2) their analytic continuation and 3) the functional equation satisfied by them.
The narrative then passes on to the abelian case of reciprocity, namely class field theory, whose history encompasses vast swathes of the history of all of number theory, more recently Gauss's theory of quadratic forms, the theory of quadratic and higher reciprocity laws studied first by Legendre, continued by Gauss, and then later by Eisenstein, Kummer, Jacobi, etc. The link between algebraic number theory and the theory of \( L \)-functions, brought to light by Dirichlet's proof of his celebrated theorem on primes in arithmetic progressions, is then used as the starting point of his discussion of class field theory -- explaining then how reciprocity is an identification of one kind of \( L \)-function (the automorphic kind) with another of quite a different kind (the ones defined by Euler products ``coming from'' Frobenius elements of Galois groups, traditionally called ``motivic'' \( L \)-functions).
The main theorems of class field theory are then stated in terms, first of \( L \)-functions, and then of abelian extensions, and finally of idèles. The abelian story of reciprocity concludes with a discussion of algebraic Hecke characters and the Brauer and Weil groups.
The next section is then devoted to a discussion of nonabelian reciprocity or nonabelian class field theory: Artin \( L \)-functions, attached to representations of the Galois groups of extensions of number fields, and the Artin conjecture regarding holomorphic continuations and functional equations of such \( L \)-functions associated with irreducible representations. Follows a description of motivic \( \zeta \)-functions: \( \zeta_A(s) \) where \( A \) is a (commutative) ring (with 1), imitating the definition of the Dedekind \( \zeta_K(s), \) with the sum now running over \textit{cofinite} ideals, i.e., those \( \mathfrak{a} \) for which \( A/\mathfrak{a} \) is finite, and then a natural generalisation of this, namely \( \zeta_X(s) \), where \( X \) is a \( \mathbb{Z} \)-scheme of finite type. Then come the \( \zeta_X(s) \) for varieties \( X \) of finite type over \( \mathbb{F}_p \), together with the well-known Riemann Hypothesis; the Hasse-Weil \( \zeta \)- and \( L \)-functions are then described. The need for Grothendieck's conjectural motives are nicely summed up in the ensuing paragraphs, and how number theorists often circumvent the difficulties of rigorously defining motives by working with compatible families of \( \ell \)-adic representations.
The Sato-Tate conjectures are then explained, and how they, in part, necessitate the introduction of the conjectural Langlands group. This is elaborated upon in the final paragraphs of this section.
Next comes a discussion of the automorphic side -- automorphic representations and the \( L \)-functions associated to them, and how they give rise to the most general form of reciprocity, the conjectured Langlands reciprocity, and the Langlands functoriality conjectures.
The last section of this interesting article is devoted to the progress towards proving the Langlands functoriality conjectures. The proof of class field theory -- abelian reciprocity --, he explains, can be thought of as consisting of two steps i) constructing the ray class fields (for which one has an explicit reciprocity law) ii) showing that any abelian extension is contained in a ray class field. Although there are other proofs of class field theory, the dominant contemporary approaches can be viewed in the above framework, and the analogue of ray class fields are the so-called Shimura varieties. Shimura varieties have been extensively studied, and a study of the cohomology of Shimura varieties have led to some of the most well-known results towards proving functoriality.
To conclude our review, the fact that someone has written such an interesting overview article is a true blessing for all those who would like to work in or to know about some of the most important problems in contemporary mathematics.
The discussions are fairly precise, yet they remain understandable even for non-experts in the area. I would fervently recommend this article to anyone who would like to work in or learn something about number theory, algebraic geometry or anything vaguely related to them.
For the entire collection see [Zbl 1472.11005].
Reviewer: Ramdin Mawia (Kolkata)Holomorphy of adjoint L-functions for \(\mathrm{GL}(n):n\le 4\)https://zbmath.org/1491.110542022-09-13T20:28:31.338867Z"Yang, Liyang"https://zbmath.org/authors/?q=ai:yang.liyangAuthor's abstract: We show entireness of complete adjoint \(L\)-functions associated to any cuspidal representations of GL(3) or GL(4) over an arbitrary number field. Twisted cases are also
investigated.
Reviewer's remarks: The author's paper is very long. It does contain explicit calculations; in order to read and enjoy the paper, the reader must be aware that it will take him or her hours to read it. The truth of some of the arguments and/or conclusions in the paper under review, does depend at several occasions on results of the author's paper quoted as [``A coarse Jacquet-Zagier trace formula for \(\mathrm{GL}(n)\) with applications'', Preprint, \url{arXiv:2003.03450}] in the References. At the time of printing of the paper under review, that paper [loc. cit.] has the status of being submitted as a preprint. Also, on page 1801, the author uses the Uchida-van der Waall Theorem (see respectively [\textit{K. Uchida}, Tohoku Math. J. (2) 27, 75--81 (1975; Zbl 0306.12007); \textit{R. W. van der Waall}, Nederl. Akad. Wet., Proc., Ser. A 78, 83--86 (1975; Zbl 0298.12003)] in which solvable Galois groups and semi-direct products of a graph with normal nontrivial abelian subgroup are involved) as well as corresponding results described in the paper by \textit{M. R. Murty} and \textit{A. Raghuram} [J. Ramanujan Math. Soc. 15, No. 4, 225--245 (2000; Zbl 1044.11099)]. Thus, when correct, somewhere in the text of the paper under review, it seems clear to the author, that the situations as described by Uchida, van der Waall, Murty and Raghuram are applicable in his work to the proof finalisation of his Theorem A. Again, due to the high amount of details of the paper, perhaps an extended seminorm will approve the results of the paper.
Reviewer: Robert W. van der Waall (Huizen)Algebraicity of the near central non-critical values of symmetric fourth \(L\)-functions for Hilbert modular formshttps://zbmath.org/1491.110552022-09-13T20:28:31.338867Z"Chen, Shih-Yu"https://zbmath.org/authors/?q=ai:chen.shih-yuLet \(\Pi\) denote a cuspidal automorphic representation of \(\mathrm{GL}_2\) over a totally real number field \(F/{\mathbb Q}\) and assume that \(\Pi\) corresponds to a cuspidal Hilbert modular form \(f_\Pi\), a normalized newform of weight \((\kappa_v)_{v\mid\infty}\). Assume that \(\kappa_v\geq 3\) for all Archimedean places \(v\) of \(F\). This is equivalent to saying that \(\Pi\) is cohomological with respect to a coefficient system \(V\) of regular highest weight.
Assume furthermore that \(\Pi\) is non-CM, i.e., it is no base change of a Hecke character in a (totally imaginary) quadratic extension of \(F\). This condition ensures that the Gelbart-Jacquet lift \(\mathrm{Sym}^2\Pi\) of \(\Pi\) to \(\mathrm{GL}_3\) is cuspidal [\textit{S. Gelbart} and \textit{H. Jacquet}, Ann. Sci. Éc. Norm. Supér. (4) 11, No. 4, 471--542 (1978; Zbl 0406.10022)]. Moreover, \(\mathrm{Sym}^2\Pi\) is cohomological as well, i.e., regular algebraic in the sense of \textit{L. Clozel} [Perspect. Math. 10, 77--159 (1990; Zbl 0705.11029)]. In particular, the finite part of \(\mathrm{Sym}^2\Pi\) is defined over the field of rationality \(\mathbb Q(\Pi)\) of \(\Pi\), which is a number field.
\textit{A. Raghuram} and \textit{F. Shahidi} [Int. Math. Res. Not. 2008, Article ID rnn077, 23 p. (2008; Zbl 1170.11009)] have shown the existence of a canonical rational structure on the Whittaker model of the finite part of \(\mathrm{Sym}^2\Pi\). Comparing this rational structure with the canonical rational structure on cuspidal cohomology of \(\mathrm{Sym}^2\Pi\) in top and bottom degrees (after a suitable choice of vectors in the Archimedean component), we obtain two well defined complex periods \(p^t(\mathrm{Sym}^2\Pi)\) and \(p^b(\mathrm{Sym}^2\Pi)\).
The author then proves the following rationality result: \[ \frac{L^{(\infty)}(1,\Pi,\mathrm{Sym}^4\otimes\omega_{\Pi}^{-2})}{\pi^{3\sum_{v\mid\infty}\kappa_v}\cdot G(\omega_\Pi)^{-3}\cdot\|f_\Pi\|\cdot p^t(\mathrm{Sym}^2\Pi)}\in\mathbb Q(\Pi). \] Moreover, the expression on the left hand side is \(\mathrm{Aut}(\mathbb C)\)-equivariant in a suitable sense. Here \(L^{(\infty)}(1,\Pi,\mathrm{Sym}^4\otimes\omega_{\Pi}^{-2})\) denotes the symmetric fourth power \(L\)-function of \(\Pi\), twisted by the negative square of the central character \(\omega_\Pi\) of \(\Pi\), \(G(\omega_\Pi)\) is a Gauss sum, and \(\|f_\Pi\|\) denotes the Petersson norm of \(f_\Pi\), defined via the Tamagawa measure on \(\mathrm{GL}_2/F\).
The main ingredients to the proof are: An explicit choice of Archimedean Whittaker function in the minimal \(\mathrm{SO}_3\)-type of \(\mathrm{Sym}^2\Pi\) due to \textit{T. Miyazaki} [Manuscr. Math. 128, No. 1, 107--135 (2009; Zbl 1158.22014)], which allows for normalizations of periods, and several rationality results for special values of the the following \(L\)-functions: the standard \(L\)-function \(L(s,\Pi)\) in the language of \textit{A. Raghuram} and \textit{N. Tanabe} [J. Ramanujan Math. Soc. 26, No. 3, 261--319 (2011; Zbl 1272.11069)]; the triple product \(L\)-function \(L(s,\Pi\times\Pi\times\Pi)\) due to \textit{P. B. Garrett} and \textit{M. Harris} [Am. J. Math. 115, No. 1, 161--240 (1993; Zbl 0776.11027)]; the Rankin-Selberg \(L\)-function \(L(s,\mathrm{Sym}^2\Pi\times \Pi)\) due to \textit{A. Raghuram} [Forum Math. 28, No. 3, 457--489 (2016; Zbl 1417.11082)]; the Rankin-Selberg \(L\)-function \(L(s, \mathrm{Sym}^2\Pi\times (\mathrm{Sym}^2\Pi)^\vee)\) by \textit{B. Balasubramanyam} and \textit{A. Raghuram} [Am. J. Math. 139, No. 3, 641--679 (2017; Zbl 06837458)]. The rationality results for the latter two \(L\)-functions are refined by the author in his Theorems 4.11 and 5.5, by completing the Archimedean period computations. These may be considered the main contribution of the paper.
Reviewer: Fabian Januszewski (Paderborn)Transfer operators and Hankel transforms between relative trace formulas. II: Rankin-Selberg theoryhttps://zbmath.org/1491.110562022-09-13T20:28:31.338867Z"Sakellaridis, Yiannis"https://zbmath.org/authors/?q=ai:sakellaridis.yiannisIn a broad sense, the idea of ``Beyond endoscopy'' in Langlands' program could be understood as comparison of trace formulas that encompasses the (stable) Arthur-Selberg trace formula as well as the relative trace formulas such as Kuznetsov trace formula: different trace formulas are to be compared via certain transfer operators. The work under review studies an enlightening example using Rankin-Selberg theory, the protagonists being the symmetric square \(L\)-function, Kuznetsov trace formula for \(\mathrm{GL}_2\) and Hankel transforms. This yields a trace formula-theoretic approach to the functional equation for \(L(\mathrm{Sym}^2)\) for \(\mathrm{GL}_2\).
Specifically, let \((V, \omega)\) be a two-dimensional symplectic space. The Rankin-Selberg variety \(\bar{X}\) is defined as \(V \times \mathrm{SL}(V)\), on which \(\tilde{G} = \mathbb{G}_m \times \mathrm{SL}(V)^2\) acts by
\[
(v, g) (a, g_1, g_2) = (avg_1, g_1^{-1} g g_2);
\]
it can also be identified with the homogeneous vector bundle \(V \overset{\mathrm{SL}(V)}{\times} \mathrm{SL}(V)^2\) over \(\mathrm{SL}(V)\) where \(\mathrm{SL}(V)\) embeds diagonally. This is an affine spherical \(\tilde{G}\)-variety, and the complement \(X\) of the zero section of the bundle \(\bar{X}\) is the open \(\tilde{G}\)-orbit. By fixing a symplectic basis for \(V\), we have
\[
X \simeq \text{diag}\bigl( \begin{smallmatrix} 1 & * \\
& 1 \end{smallmatrix} \bigr) \big\backslash \mathrm{SL}(2)^2.
\]
The strategy is ultimately based on a space of non-standard test measures \(\mathcal{S}(\bar{X} / \mathrm{SL}_2)^\circ\) over local fields and its endomorphism \(\mathcal{H}_X^\circ\). In Section 8, one defines \(\mathcal{S}(\bar{X} / \mathrm{SL}_2)^\circ\) over a local field. It lies between \(\mathcal{S}(X / \mathrm{SL}_2)\) (the naive Schwartz space) and \(\mathcal{S}(\bar{X} / \mathrm{SL}_2)\) (which produces the \(L\)-values \(L(\mathrm{Sym}^2)\) and \(L(\mathrm{triv})\) in relative trace formula). It is designed to yield the \(L\)-value \(L(\mathrm{Sym}^2)\) only. These spaces are described in terms of Mellin transforms, and the relevant tools for this are given in Section 7. Fix an additive character \(\psi\). The endomorphism \(\mathcal{H}_X^\circ\) is a factor of the fiberwise Fourier transform on \(\bar{X}\): it produces the symmetric square \(\gamma\)-factor when applied to relative characters (Theorem 8.3.5). To describe such Fourier transforms, it is best to consider spaces of half-densities \(\mathcal{D}(\cdots)\) instead of measures (i.e.,\ \(1\)-densities), although it is routine to pass between them.
In Section 9, these spaces are used to construct a space \(\mathcal{S}^-_{L(\mathrm{Sym}^2, 1)}(N, \psi \backslash G / N, \psi)\) of test measures for the Kuznetsov trace formula for \(G = \mathbb{G}_m \times \mathrm{SL}_2\). The basic tool is the unfolding map \(\mathcal{U}\) from \(\bar{X}\) to \(N, \psi \backslash \tilde{G}\), a sophisticated paraphrase of the unfolding technique in Rankin-Selberg theory. One also obtains a Hankel transform
\[
\mathcal{H}_{\mathrm{Sym}^2}: \mathcal{D}^-_{L(\mathrm{Sym}^2, 1/2)}(N, \psi \backslash G / N, \psi) \xrightarrow{\sim} \mathcal{D}^-_{L(\mathrm{Sym}^{2, \vee}, 1/2)}(N, \psi \backslash G / N, \psi).
\]
Its effect on the relative character of \(\pi\) is the multiplication by \(\gamma(\pi, \mathrm{Sym}^2, \frac{1}{2}, \psi)\), when the irreducible generic representation \(\pi\) varies in a family \(\pi \otimes |\cdot|^s\); see Theorem 9.0.1.
Following the general philosophy of comparison of trace formula, all these spaces contain a ``basic vector'' in the unramified setting, preserved by Hankel transforms, and the relation between unramified Hecke action and Hankel transforms has a transparent description. Furthermore, \(\mathcal{H}_{\mathrm{Sym}^2}\) is compatible with boundary degeneration (Theorem 9.6.1). Such phenomena also pertain to the Hankel transform \(\mathcal{H}_{\mathrm{Std}}\) for the standard \(L\)-factor considered by Jacquet, and suggest that Hankel transform is a deformation of its abelian analogue.
In Section 10, one constructs a transfer operator \(\mathcal{T}_T\) from \(\mathcal{S}^-_{L(\mathrm{Sym}^2, 1)}(N, \psi \backslash G /N, \psi)\) to \(\mathcal{S}(T)\) with many properties (Theorem 10.1.1), where \(T\) is the torus associated with some quadratic (possibly split) extension of the base field \(F\). It is constructed by first passing to \(\kappa\)-orbital integrals on \(\mathrm{SL}_2\) (Theorem 10.3.2) and then to \(T\) by Labesse-Langlands theory, suggesting that beyond endoscopy is not entirely disjoint from endoscopy. As an application, one obtains a simple expression for the stable characters on \(\mathrm{SL}_2\) lifted from \(T\), known as Gelfand-Graev-Piatetski-Shapiro formula (see 10.29). The relation to Venkatesh's thesis is discussed at length in 10.1.
Reviewer: Wen-Wei Li (Beijing)Analytic Lie extensions of number fields with cyclic fixed points and tame ramificationhttps://zbmath.org/1491.110572022-09-13T20:28:31.338867Z"Hajir, Farshid"https://zbmath.org/authors/?q=ai:hajir.farshid"Maire, Christian"https://zbmath.org/authors/?q=ai:maire.christianLet \(\Gamma\) be a pro-\(p\) group. If it is torsion-free and with \(\varepsilon=1\) for \(p>2\), \(\varepsilon=2\) for \(p=2\) one has \([\Gamma,\Gamma]\subset\Gamma^{p^\varepsilon}\), then \(\Gamma\) is called \textit{uniform}. The authors consider the following version of the Tame Fontaine-Mazur Conjecture [\textit{J.-M. Fontaine} and \textit{B. Mazur}, in: Elliptic curves, modular forms, \& Fermat's last theorem. Proceedings ot the conference on elliptic curves and modular forms held at the Chinese University of Hong Kong, December 18-21, 1993. Cambridge, MA: International Press. 41--78 (1995; Zbl 0839.14011)]:
If \(K\) is an algebraic number field and \(\Gamma\) is a uniform pro-\(p\) group of dimension \(d\ge3\), then there is no finitely and tamely ramified Galois extension \(L/K\) having \(\Gamma\) for its Galois group.
In the case of unramified extensions some evidence for this conjecture has been given by \textit{N. Boston} [J. Number Theory 42, No. 3, 285--291 (1992; Zbl 0768.11044); J. Number Theory 75, No. 2, 161--169 (1999; Zbl 0928.11050)] and \textit{C. Maire} [Math. Res. Lett. 14, No. 4, 673--680 (2007; Zbl 1188.11060)]. In particular it has been shown by Boston that if \(K/k\) is a cyclic extension of prime order \(\ell\mid p-1\) and \(p\nmid h(k)\),then there is no non-trivial unramified extension of \(L/K\) such that \(L/k\) is Galois and Gal\((L/K)\) is a uniform pro-\(p\) group of finite dimension.
The authors present an extension of Boston's approach to the case when \(L/K\) is tamely ramified. They assume that \(K\) has a subfield \(k\) with \([K:k]=\ell\), a prime \(\ne p\) with \(L/k\) being Galois and write in the abstract:
``Letting \(\sigma\) be a generator of Gal\((K/k)\), we study the constraints posed on the arithmetic of \(L/K\) by the cyclic action of \(\sigma\) on \(\Gamma\), focusing on the cri\-ti\-cal role played by the fixed points of this action, and their relation to the ramification in \(L/K\). The method of Boston works only when there are no non-trivial fixed points for this action. We show that even in the presence of arbitrarily many fixed points, the action of \(\sigma\) places severe arithmetic conditions on the existence of finitely and tamely ramified uniform \(p\)-adic analytic extensions over \(K\), which in some instances leads us to be able to deduce the non-existence of such extensions over \(K\) from their non-existence over \(k\).''
Reviewer: Władysław Narkiewicz (Wrocław)Cyclotomic torsion points in elliptic schemeshttps://zbmath.org/1491.110582022-09-13T20:28:31.338867Z"Giacomini, Michele"https://zbmath.org/authors/?q=ai:giacomini.micheleLet \(E\) be an elliptic curve defined over a number field \(k\). By a work of Serre about Galois representations attached to elliptic curves, there are only a finitely many torsion points of \(E\) which are defined over the cyclotomic closure \(k^{\mathrm{c}}\). The author of the paper under review proves the following for an elliptic scheme: Let \(\mathcal E/C\) be an elliptic scheme over a curve \(C\). Let \(s : C\to \mathcal E\) be a section which is not identically torsion. Let \(f : C \to \mathbb A^1\) be a non constant rational function, and all are defined over \(k\). If \(n\) is a natural number, then there are only finitely many points \(P\in C\) such that \(f(P)\) is the sum of \(n\) roots of unity and such that \(s(P)\in \mathcal E_P\) is torsion.
The author believes that the restriction on the length \(n\) is a technical condition, and that the result is true in general without the restriction. The author follows the methods used by Masser, Pila, and Zannier to prove the Manin-Mumford conjecture and the relative Manin-Mumford conjecture, but he makes further contributions in this work for handling the presence of semi-algebraic sets of dim \(\ge 1\) in the transcendental variety defined using logarithms.
Reviewer: Sungkon Chang (Savannah)Rank growth of elliptic curves in non-abelian extensionshttps://zbmath.org/1491.110592022-09-13T20:28:31.338867Z"Lemke Oliver, Robert J."https://zbmath.org/authors/?q=ai:lemke-oliver.robert-j"Thorne, Frank"https://zbmath.org/authors/?q=ai:thorne.frankIn the paper under review, the authors prove the following result. Let \(K\) be a number field with \([K:{\mathbb Q}]=d\geq 2\), \(\tilde K\) its Galois closure and \(\mbox{Gal} (\tilde K/{\mathbb Q} \cong S_d\). Denote by \({\mathcal F}_d (X)\) the set of all such \(K\) with \(|\mbox{Disc} (K) | \leq X\) for a given \(X\). For a given elliptic curve \(E/{\mathbb Q}\) there is a constant \(c_d > 0\) such that for each \(\varepsilon = \pm 1\), the number of \(K \in \mathcal F_d (X)\) with \(\mbox{rk } (E/K) > \mbox{rk } (E/{\mathbb Q})\) and root number \(w (E_K)=\varepsilon\) is \(\gg_{E, d, \varepsilon} X^{c_d - \varepsilon}\). Moreover, \(c_d > 1/4 - \varepsilon\) as \(d \to \infty\). This implies that there are at least \(X^{c_d-\varepsilon}\) such fields where the rank grows. Subject to the parity conjecture, the authors obtain the same result for fields for which the rank grows by at least two.
Reviewer: Tanush Shaska (Vlorë)Commensurability in Mordell-Weil groups of abelian varieties and torihttps://zbmath.org/1491.110602022-09-13T20:28:31.338867Z"Banaszak, Grzegorz"https://zbmath.org/authors/?q=ai:banaszak.grzegorz"Blinkiewicz, Dorota"https://zbmath.org/authors/?q=ai:blinkiewicz.dorotaFrom the text: We investigate local to global properties for commensurability in Mordell-Weil groups of abelian varieties and tori via reduction maps.
In more detail, local to global properties for detecting linear relations in Mordell-Weil groups
of abelian varieties and tori have been investigated by numerous authors. Commensurability
questions in the Mordell-Weil groups have not yet been investigated in relation to reduction maps. In this paper we establish the relations between local to global detecting properties and local to global commensurability properties. We apply these results to Mordell-Weil groups of abelian varieties and tori. The structure of the paper is as follows. At the end of this introduction we define local to global commensurability properties. We also define notion of strong commensurability
in abelian groups with finite torsion. Then we define local to global properties for strong commensurability. In section 2 we investigate relations between local to global commensurability properties and local to global detecting properties. In section 3 we give examples of classes of abelian varieties and tori where the local to global strong commensurability property holds. In both cases we show examples of classes of abelian varieties and tori where the criterion fails.
As a corollary we obtain, in each case, four different Deligne 1-motives over a ring of integers, which become all equal to a torsion 1-motive, after base change and application of reduction map for almost all residue fields. In section 4 we give examples where one can check the strong commensurability in Mordell-Weil groups of abelian varieties and tori by finite number of reductions.Purely additive reduction of abelian varieties with torsionhttps://zbmath.org/1491.110612022-09-13T20:28:31.338867Z"Melistas, Mentzelos"https://zbmath.org/authors/?q=ai:melistas.mentzelosSummary: Let \(\mathcal{O}_K\) be a discrete valuation ring with fraction field \(K\) of characteristic 0 and algebraically closed residue field \(k\) of characteristic \(p > 0\). Let \(A / K\) be an abelian variety of dimension \(g\) with a \(K\)-rational point of order \(p\). In this article, we are interested in the reduction properties that \(A / K\) can have. After discussing the general case, we specialize to \(g = 1\), and we study the possible Kodaira types that can occur.Cycles in the de Rham cohomology of abelian varieties over number fieldshttps://zbmath.org/1491.110622022-09-13T20:28:31.338867Z"Tang, Yunqing"https://zbmath.org/authors/?q=ai:tang.yunqingSummary: In his 1982 paper, \textit{A. Ogus} [Lect. Notes Math. 900, 357--414 (1982; Zbl 0538.14010)] defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of \(\ell\)-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus' prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings' isogeny theorem due to Bost and the known cases of the Mumford-Tate conjecture.On a conjecture of Pappas and Rapoport about the standard local model for \(\mathrm{GL}_d\)https://zbmath.org/1491.110632022-09-13T20:28:31.338867Z"Muthiah, Dinakar"https://zbmath.org/authors/?q=ai:muthiah.dinakar"Weekes, Alex"https://zbmath.org/authors/?q=ai:weekes.alex"Yacobi, Oded"https://zbmath.org/authors/?q=ai:yacobi.odedLet \(n\) and \(e\) be integers greater than or equal to \(2\). Pappas and Rapoport conjectured that the subscheme \[\mathcal{N}_{n,e} = \{ A \in \mathrm{Mat}_{n \times n} \: : \: A^e = 0, \det(\lambda-A) = \lambda^n \}\] of the scheme \(\mathrm{Mat}_{n \times n}\) of \(n \times n\) matrices is reduced (Conjecture 5.8 of [\textit{G. Pappas} and \textit{M. Rapaport}, J. Algebraic Geom 12, 107--145 (2003; Zbl 1063.14029)]). They showed that this conjecture implies the flatness of the standard model of \(\mathrm{GL}_d\) in certain situations.
The above conjecture is proved in full generality in this paper.
Reviewer: Salman Abdulali (Greenville)On the existence of curves with prescribed \(a\)-numberhttps://zbmath.org/1491.110642022-09-13T20:28:31.338867Z"Zhou, Zijian"https://zbmath.org/authors/?q=ai:zhou.zijianLet \(k\) be an algebraically closed field of characteristic \(p>0\), let \(X\) be a curve defined over \(k\), and let \(\text{Jac}(X)\) be its Jacobian. One of the most important invariants of \(X\) is its \(a\)-number \(a_{X}\), which is defined by \[a_{X}=\text{dim}_{k}(\text{Hom}(\alpha_{p}, \text{Jac}(X))),\] where \(\alpha_{p}\) is the group scheme which is the kernel of Frobenius on the additive group scheme \(\mathbb{G}_{a}\). It is well known that the \(a\)-number of \(X\) is equal to \(g-r\), where \(g\) is the genus of \(X\) and \(r\) is the rank of the Cartier-Manin matrix, that is, the matrix for the Cartier operator defined on \(H^{0}(X,\Omega^{1}_{X})\).
In this paper, the author studies the existence of Artin-Schreier curves with large \(a\)-number, namely \(a_{X}=g-1\) and \(a_{X}=g-2\), proving among other important results that such curves can be written in some particular forms. Also, by computing the rank of the Hasse-Witt matrix of the curve, bounds on the \(a\)-number of trigonal curves of genus \(5\) in small characteristic are also given.
Reviewer: Mariana Coutinho (São Carlos)Explicit calculation of the mod 4 Galois representation associated with the Fermat quartichttps://zbmath.org/1491.110652022-09-13T20:28:31.338867Z"Ishitsuka, Yasuhiro"https://zbmath.org/authors/?q=ai:ishitsuka.yasuhiro"Ito, Tetsushi"https://zbmath.org/authors/?q=ai:ito.tetsushi"Ohshita, Tatsuya"https://zbmath.org/authors/?q=ai:ohshita.tatsuyaVisibility and the Birch and Swinnerton-Dyer conjecture for analytic rank zerohttps://zbmath.org/1491.110662022-09-13T20:28:31.338867Z"Agashe, Amod"https://zbmath.org/authors/?q=ai:agashe.amodFrom the abstract: ``Let \(E\) be an optimal elliptic curve over \(\mathbb{Q}\) of conductor \(N\) having analytic rank zero, i.e., such that the \(L\)-function \(L_E(s)\) of \(E\) does not vanish at \(s = 1\). Suppose there is another optimal elliptic curve over \(\mathbb{Q}\) of the same conductor \(N\) whose Mordell-Weil rank is greater than zero and whose associated newform is congruent to the newform associated to \(E\) modulo a power \(r\) of a prime \(p\). The theory of visibility then shows that under certain additional hypotheses involving \(p, r\) divides the product of the order of the Shafarevich-Tate group of \(E\) and the orders of the arithmetic component groups of \(E\). We extract an explicit integer factor from the Birch and Swinnerton-Dyer conjectural formula for the product mentioned above, and under some hypotheses similar to the ones made in the situation above, we show that \(r\) divides this integer factor. This provides theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank zero case.''
The author treats the case of the analytic rank zero while he previously obtained the case of the analytic rank one in [Int. Math. Res. Not. 2009, No. 15, 2899--2913 (2009; Zbl 1183.11035)] but the methods of two articles are different. One is the homology groups and the other is the Galois cohomology groups. Further, this article generalizes the case of \(r=p\) obtained in [the author, J. Reine Angew. Math. 644, 159--187 (2010; Zbl 1221.11147)].
Reviewer: Kazuma Morita (Sapporo)Sets of exact approximation order by complex rational numbershttps://zbmath.org/1491.110672022-09-13T20:28:31.338867Z"He, YuBin"https://zbmath.org/authors/?q=ai:he.yubin"Xiong, Ying"https://zbmath.org/authors/?q=ai:xiong.yingFor a function \(\Psi:(0,\infty)\to (0,\infty)\) denote by Exac\(_{\mathbb R}(\Psi)\) the set of real numbers that are approximable to order \(\Psi\) exactly. \textit{V. Jarnik} [Math. Z. 33, 505--543 (1931; JFM 57.1370.01)] used the theory of continued fractions to show that Exac\(_{\mathbb R}(\Psi)\) is uncountable if \(\Psi\) is non-increasing and satisfies \(\Psi(x) = o(x^{-2})\). His method provides no information regarding Hausdorff dimension Exac\(_{\mathbb R}(\Psi)\).\par \textit{Y. Bugeaud} [Math. Ann. 327, No. 1, 171--190 (2003; Zbl 1044.11059)], \textit{Y. Bugeaud} [Unif. Distrib. Theory 3, No. 2, 9--20 (2008; Zbl 1212.11071)] and \textit{Y. Bugeaud} and \textit{C. G. Moreira} [Acta Arith. 146, No. 2, 177--193 (2011; Zbl 1211.11084)] determined the Hausdorff dimension of Exac\(_{\mathbb R}(\Psi)\) for functions \(\Psi\) that are non-increasing and satisfy \(\Psi(x) = o(x^{-2})\). The situation becomes more subtle if we remove the assumption \(\Psi(x) = o(x^{-2})\).\par In this paper, the authors consider the analogous problem for complex numbers. If \(\Psi\) is non-increasing and \(\Psi(x) = o(x^{-2})\), then the authors determine the Hausdorff dimension and the packing dimension of Exact\((\Psi)\). They also give estimates for the Hausdorff dimension of Exact\((\Psi)\) without the condition \(\Psi(x) = o(x^{-2})\).
Reviewer: István Gaál (Debrecen)Dirichlet is not just bad and singularhttps://zbmath.org/1491.110682022-09-13T20:28:31.338867Z"Beresnevich, Victor"https://zbmath.org/authors/?q=ai:beresnevich.victor-v"Guan, Lifan"https://zbmath.org/authors/?q=ai:guan.lifan"Marnat, Antoine"https://zbmath.org/authors/?q=ai:marnat.antoine"Ramírez, Felipe"https://zbmath.org/authors/?q=ai:ramirez.felipe-a"Velani, Sanju"https://zbmath.org/authors/?q=ai:velani.sanju-lAuthors' abstract: It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of \textit{H. Davenport} and \textit{W. Schmidt} [Sympos. Math., Roma 4, Teoria Numeri, Dic. 1968, e Algebra, Marzo 1969, 113--132 (1970; Zbl 0226.10032)] which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main ``continuum'' result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dirichlet improvable points.
Reviewer: István Gaál (Debrecen)On the distribution of \(\alpha p\) modulo one in quadratic number fieldshttps://zbmath.org/1491.110692022-09-13T20:28:31.338867Z"Baier, Stephan"https://zbmath.org/authors/?q=ai:baier.stephan"Mazumder, Dwaipayan"https://zbmath.org/authors/?q=ai:mazumder.dwaipayan"Technau, Marc"https://zbmath.org/authors/?q=ai:technau.marcAuthors' abstract: We investigate the distribution of \(\alpha p\) modulo one in quadratic number fields \(\mathbb{K}\) with class number one, where \(p\) is restricted to prime elements in the ring of integers of \(\mathbb{K} \). Here we improve the relevant exponent 1/4 obtained by the first- and third-named authors for imaginary quadratic number fields [J. Théor. Nombres Bordx. 32, No. 3, 719--760 (2020; Zbl 1459.11148)] and by the first- and second-named authors for real quadratic number fields [Math. Z. 299, No. 1--2, 699--750 (2021; Zbl 1484.11155)] to 7/22. This generalizes a result of \textit{G. Harman} [Q. J. Math. 70, No. 4, 1505--1519 (2019; Zbl 1469.11235)] who obtained the same exponent 7/22 for \(\mathbb{Q} (i)\) by extending his method which gave this exponent for \(\mathbb{Q} \) [\textit{G. Harman}, Proc. Lond. Math. Soc. (3) 72, No. 2, 241--260 (1996; Zbl 0874.11052)]. Our proof is based on an extension of Harman's sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in the above-mentioned work by the first- and second-named authors, for which we use analytic properties of Hecke \(\mathrm{L}\)-functions with Größencharacters.
Reviewer: Giovanni Coppola (Napoli)Density of sequences of the form \(x_n=f(n)^n\) in \([0,1]\)https://zbmath.org/1491.110702022-09-13T20:28:31.338867Z"Saunders, J. C."https://zbmath.org/authors/?q=ai:saunders.j-cIn this paper, the author shows that certain sequences $x_n$ satisfying $\log x_n=ng(n)$, where $g$ is a real analytic function on the interval $[-1,1]$ are dense subsets of the interval $[0,1]$. The proofs make use of Diophantine approximation type arguments, continued fractions and an adaptation of Zaharescu's breakthrough on the small values of square multiples modulo 1.
Reviewer: Emre Alkan (İstanbul)The Okada space and vanishing of \(L(1,f)\)https://zbmath.org/1491.110712022-09-13T20:28:31.338867Z"Murty, M. Ram"https://zbmath.org/authors/?q=ai:murty.maruti-ram"Pathak, Siddhi"https://zbmath.org/authors/?q=ai:pathak.siddhi-sFix a positive integer \(N\ge 2\). Following Chowla, the authors associate the \(L\)-series \(L(s,f):=\sum_{n\ge1}\frac{f(n)}{n^s}\) to each function \(f:\mathbb Z\to\mathbb C\) with period \(N\). Using a characterization derived by Okada for the vanishing of \(L(1,f)\), the authors construct an explicit basis for the \(\mathbb Q\)-vector space, \[\mathcal O(N) = \{f \bmod N : f(n)\in \mathbb Q, L(1,f)=0\}.\] The authors analyze the structure of this space and use the explicit basis to extend earlier works of Baker-Birch-Wirsing and Murty-Saradha. The arithmetical nature of Euler's constant~\(\gamma\) emerges as a central question in these extensions.
Reviewer: Jan Šustek (Ostrava)Arithmetic properties of Euler-type series with a Liouvillian polyadic parameterhttps://zbmath.org/1491.110722022-09-13T20:28:31.338867Z"Chirskii, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevichSummary: This paper states that, for any nonzero linear form \(h_0 f_0(1) + h_1f_1(1)\) with integer coefficients \(h_0\), \(h_1\), there exist infinitely many \(p\)-adic fields where this form does not vanish. Here, \(f_0(1) = \mathop \sum \limits_{n = 0}^\infty{{\left( \lambda \right)}_n}\) and \(f_1\left( 1 \right) = \mathop \sum \limits_{n = 0}^\infty{{\left( {\lambda + 1} \right)}_n} \), where \(\lambda\) is a Liouvillian polyadic number and \(( \lambda)_n\) stands for the Pochhammer symbol. This result shows the possibility of studying the arithmetic properties of values of hypergeometric series with transcendental parameters.Uniform distribution of the weighted sum-of-digits functionshttps://zbmath.org/1491.110732022-09-13T20:28:31.338867Z"Mišík, Ladislav"https://zbmath.org/authors/?q=ai:misik.ladislav-jun"Porubský, Štefan"https://zbmath.org/authors/?q=ai:porubsky.stefan"Strauch, Oto"https://zbmath.org/authors/?q=ai:strauch.otoOne can begin with authors' abstract:
``The higher-dimensional generalization of the weighted \(q\)-adic sum-of-digits functions \(s_{q, \gamma}(n)\), \(n =0, 1, 2,\dots,\) covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., \(d\)-dimensional van der Corput-Halton or \(d\)-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted \(q\)-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function \(g(x)= x\) implies the uniform distribution modulo one of the weighted \(q\)-adic sum-of-digits function \(s_{q, \gamma}(n)\), \(n =0, 1, 2,\dots,\). We also prove the uniform distribution modulo one of related sequences \(h_{1 s_{q, \gamma}} (n)+h_{2 s_{q, \gamma}} (n +1)\), where \(h_1\) and \(h_2\) are integers such that \(h_1 + h_2 \neq 0\) and that the akin two-dimensional sequence \((s_{q, \gamma} (n), s_{q, \gamma} (n +1))\) cannot be uniformly distributed modulo one if \(q \geq 3\). The properties of the two-dimensional sequence \((s_{q, \gamma} (n), s_{q, \gamma} (n +1))\), \(n=0,1,2,\dots,\) will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.''
The special attention is given to explanations of the main notions and to examples of certain sequences. For example, uniformly distributed mod \(1\) \(d\)-dimensional vectors and the weighted \(q\)-adic sum-of-digits function, the (extreme) discrepancy of a finite sequence and an asymptotic distribution function, the \(d\)-dimensional van der Corput-Halton and Halton-Kronecker sequences, as well as the \(d\)-dimensional Kronecker sequence, etc., are noted. Also, Pillichshammer's problem is considered.
The main and auxiliary results are proven with explanations. The main techniques of proofs are described.
Reviewer: Symon Serbenyuk (Kyïv)On Schneider's continued fraction map on a complete non-Archimedean fieldhttps://zbmath.org/1491.110742022-09-13T20:28:31.338867Z"Haddley, A."https://zbmath.org/authors/?q=ai:haddley.alena"Nair, R."https://zbmath.org/authors/?q=ai:nair.rahul|nair.reshmi|nair.ranjit|nair.ravi|nair.ravindraThe present paper deals with Schneider's continued fraction map and its entropy. One can note the following description of this research:
``Let \({\mathcal{M}}\) denote the maximal ideal of the ring of integers of a non-Archimedean field \(K\) with residue class field \(k\) whose invertible elements, we denote \(k^{\times}\), and a uniformizer we denote \(\pi \). In this paper, we consider the map \(T_v: {\mathcal{M}} \rightarrow{\mathcal{M}}\) defined by
\[
T_v(x) = \frac{\pi^{v(x)}}{x} - b(x),
\]
where \(b(x)\) denotes the equivalence class to which \(\frac{\pi^{v(x)}}{x}\) belongs in \(k^{\times}\). We show that \(T_v\) preserves Haar measure \(\mu\) on the compact abelian topological group \({\mathcal{M}} \). Let \({\mathcal{B}}\) denote the Haar \(\sigma \)-algebra on \({\mathcal{M}} \). We show the natural extension of the dynamical system \(({\mathcal{M}}, {\mathcal{B}}, \mu, T_v)\) is Bernoulli and has entropy \(\frac{\#( k)}{\#( k^{\times})}\log (\#( k))\). The first of these two properties is used to study the average behaviour of the convergents arising from \(T_v\). Here for a finite set \(A\) its cardinality has been denoted by \(\# (A)\). In the case \(K = {\mathbb{Q}}_p\), i.e. the field of \(p\)-adic numbers, the map \(T_v\) reduces to the well-studied continued fraction map due to \textit{Th. Schneider} [in: Sympos. Math., Roma 4, Teoria Numeri, Dic. 1968, e Algebra, Marzo 1969, 181--189 (1970; Zbl 0222.10035)].''
Special attention is given to topological fields. The notion of non-Archimedean fields is explained. The field of \(p\)-adic numbers and the field of formal Laurent series in finite characteristic are considered. It is noted that only two types of non-Archimedean local fields there are finite extensions of the field of \(p\)-adic numbers for some rational prime \(p\) and the field of formal Laurent series over a finite field. Auxiliary references are given.
The notions of Schneider's continued fraction map and Schneider's continued fraction expansions are explained. The measure-theoretic entropy is considered. Some applications of Schneider's continued fractions are described.
Reviewer: Symon Serbenyuk (Kyïv)Multilinear exponential sums with a general class of weightshttps://zbmath.org/1491.110752022-09-13T20:28:31.338867Z"Kerr, Bryce"https://zbmath.org/authors/?q=ai:kerr.bryce"Macourt, Simon"https://zbmath.org/authors/?q=ai:macourt.simonIn the paper under review, the authors study weighted multilinear exponential sums \[ S(\mathcal{X}_1,\dots,\mathcal{X}_n;\omega_1,\dots,\omega_n):=\sum_{x_1\in \mathcal{X}_1}\cdots\sum_{x_n\in \mathcal{X}_n}\omega_1(\mathbf{x})\cdots\omega_n(\mathbf{x}) e_p(x_1x_2\cdots x_n),\] where \(\mathcal{X}_1,\dots,\mathcal{X}_n\) are subsets of \(\mathbb{F}_p^*\), \(e_p(\theta)=\exp(2\pi i\theta /p)\) with \(p\) a prime.
\textit{J. Bourgain} [Geom. Funct. Anal. 18, No. 5, 1477--1502 (2009; Zbl 1162.11043)] showed that if \(|\mathcal{X}_i|>p^{\varepsilon}\) and \(|\mathcal{X}_1|\cdots|\mathcal{X}_n|\geq p^{1+\varepsilon}\) then \[ S(\mathcal{X}_1,\dots,\mathcal{X}_n;1,\dots,1)\ll p^{-\delta}|\mathcal{X}_1|\cdots|\mathcal{X}_n|\] with \(\delta>0\) depending on \(\varepsilon\).
The main result of the authors is a bound on the general weighted sum \(S(\mathcal{X}_1,\dots,\mathcal{X}_n;\omega_1,\dots,\omega_n)\). The expression is too complicated to restate here -- the authors give various examples where the bound is nontrivial (mostly depending on the sizes of the sets \(\mathcal{X}_i\)). The method of proof uses the recent improvement due to \textit{I. D. Shkredov} [Trans. Mosc. Math. Soc. 2018, 231--281 (2018; Zbl 1473.11034); translation from Tr. Mosk. Mat. O.-va 79, No. 2, 271--334 (2018)] of Bourgain's result which is based on geometric incidence estimates of \textit{M. Rudnev} [Combinatorica 38, No. 1, 219--254 (2018; Zbl 1413.51001)].
Reviewer: Thomas Stoll (Vandœuvre-lès Nancy)Arrows of times, non integer operators, self-similar structures, zeta functions and Riemann hypothesis: a synthetic categorical approachhttps://zbmath.org/1491.110762022-09-13T20:28:31.338867Z"Alain, Le Méhauté"https://zbmath.org/authors/?q=ai:le-mehaute.a-j-y"Philippe, Riot"https://zbmath.org/authors/?q=ai:philippe.riotSummary: The authors have previously reported the existence of a morphism between the Riemann zeta function and the ``Cole and Cole'' canonical transfer functions observed in dielectric relaxation, electrochemistry, mechanics and electromagnetism. The link with self-similar structures has been addressed for a long time and likewise the discovered of the incompleteness which may be attached to any dynamics controlled by non-integer derivative operators. Furthermore it was already shown that the Riemann Hypothesis can be associated with a transition of an order parameter given by the geometric phase attached to the fractional operators. The aim of this note is to show that all these properties have a generic basis in category theory. The highlighting of the incompleteness of non-integer operators considered as critical by some authors is relevant, but the use of the morphism with zeta function reduces the operational impact of this issue without limited its epistemological consequences.On the maximum of cotangent sums related to the Riemann hypothesis in rational numbers in short intervalshttps://zbmath.org/1491.110772022-09-13T20:28:31.338867Z"Maier, Helmut"https://zbmath.org/authors/?q=ai:maier.helmut"Rassias, Michael Th."https://zbmath.org/authors/?q=ai:rassias.michael-thLet
\[
c_0\left(\frac{r}{b}\right)=-\sum_{m=1}^{b-1}\frac{m}{b}\cot\left(\frac{\pi mr}{b}\right).
\]
In this paper the authors obtain lower bounds for \(\max|c_0\left(\frac{r}{b}\right)|\) where (i) the numerator \(r\) is restricted to the sequence of prime numbers, and (ii) fractions \(\frac{r}{b}\) simultaneously varying the numerator \(r\) and the denominator \(b\).
Reviewer: Mehdi Hassani (Zanjan)On shuffle-type functional relations of desingularized multiple zeta-functionshttps://zbmath.org/1491.110782022-09-13T20:28:31.338867Z"Komiyama, Nao"https://zbmath.org/authors/?q=ai:komiyama.naoThe multiple zeta function (of Euler-Zagier type) is defined by
\[
\zeta(s_1,\ldots,s_r) =\sum_{0<m_1<\cdots <m_r}\frac{1}{m_1^{s_1}\cdots m_r^{s_r}},
\]
which converges absolutely in the region
\[
\{(s_1,\ldots,s_r)\in\mathbb{C}^r\,|\,\Re(s_{r-k+1}+\cdots+s_r)>k\ (1\le k\le r)\}.
\]
It is known that \(\zeta(s_1,\ldots,s_r)\) can be continued meromorphically to \(\mathbb{C}^r\) with singularities at
\begin{align*}
s_r&=1,\\
s_{r-1}+s_r&=2,1,0,-2,-4,\ldots,\\
s_{r-k+1}+\cdots+s_r&=k-n \ \ (3\le k\le r,\ n\in\mathbb{Z}_{\ge 0}).
\end{align*}
In [Am. J. Math. 139, No. 1, 147--173 (2017; Zbl 1369.11065)], to study values at nonpositive integer points of \(\zeta(s_1,\ldots,s_r)\) ``rigorously'', \textit{H. Furusho} et al. introduced the \textit{desingularized multiple zeta function} \(\zeta^{\text{des}}_r(s_1,\ldots,s_r)\) which is entire in \(\mathbb{C}^r\) and can be expressed as a finite ``linear'' combination of shifted multiple zeta functions.
In this paper, the author obtains the following shuffle-type product formulas of desingularized multiple zeta function: For \(s_1,\ldots,s_p\in\mathbb{C}\) and \(l_1,\ldots,l_q\in\mathbb{Z}_{\ge 0}\), it holds that
\begin{align*}
& \zeta^{\text{des}}_p(s_1,\ldots,s_p)\zeta^{\text{des}}_q(-l_1,\ldots,-l_q)\\
&=\sum_{\substack{i_b+j_b=l_b \\
i_b,j_b\ge 0 \\
1\le b\le q}}\prod^{q}_{a=1}(-1)^{i_a}\binom{l_a}{i_a} \zeta^{\text{des}}_{p+q}(s_1,\ldots,s_{p-1},s_p-i_1-\cdots-i_q,-j_1,\ldots,-j_q).
\end{align*}
This gives a generalization of author's previous results obtained in [RIMS Kôkyûroku Bessatsu B83, 83--104 (2020; Zbl 1461.11117)].
Reviewer: Yoshinori Yamasaki (Matsuyama)Joint discrete approximation of analytic functions by Hurwitz zeta-functionshttps://zbmath.org/1491.110792022-09-13T20:28:31.338867Z"Balčiūnas, Aidas"https://zbmath.org/authors/?q=ai:balciunas.aidas"Garbaliauskienė, Virginija"https://zbmath.org/authors/?q=ai:garbaliauskiene.virginija"Lukšienė, Violeta"https://zbmath.org/authors/?q=ai:luksiene.violeta"Macaitienė, Renata"https://zbmath.org/authors/?q=ai:macaitiene.renata"Rimkevičienė, Audronė"https://zbmath.org/authors/?q=ai:rimkeviciene.audroneLet
\[
D=\{s\in\mathbb{C}:1/2<\Re(s)<1\},
\]
and \(H(D)\) be the space of analytic functions over \(D\), equipped with the topology induced by uniform convergence on compact subsets.
The authors present a result regarding the approximation of functions in \(H^r(D)\) by the Hurwitz zeta function
\[
\zeta(s, \alpha)=\sum_{m=0}^{\infty} \frac{1}{(m+\alpha)^{s}}.
\]
The statement is as follows.
Take the numbers \(0<\alpha_{j}<1, \alpha_{j} \neq 1 / 2\) and \(0<h_{j}, j=1, \ldots, r\) arbitrarily. Then there exists a non-empty, closed set \(F_{{\underline{\alpha}}, \underline{h}}\) of \(H^{r}(D)\) such that, for every compact sets \(K_{1}, \ldots, K_{r}\) in \(D\), and for any \(\left(f_{1}, \ldots, f_{r}\right) \in F_{\underline{\alpha}, \underline{h}}\),
\[
\liminf _{N \rightarrow \infty} \frac{1}{N+1} \#\left\{0 \leqslant k \leqslant N: \sup _{1 \leqslant j \leqslant r \leqslant K_{j}} \sup _{s \in K_{j}}\left|\zeta\left(s+i k h_{j}, \alpha_{j}\right)-f_{j}(s)\right|<\varepsilon\right\}>0
\]
holds for every \(\varepsilon>0\).
It is also shown, that liminf can be replaced by lim for all but at most countably many \(\varepsilon>0\).
Reviewer: István Mező (Nanjing)Extension of the four Euler sums being linear with parameters and series involving the zeta functionshttps://zbmath.org/1491.110802022-09-13T20:28:31.338867Z"Sofo, Anthony"https://zbmath.org/authors/?q=ai:sofo.anthony"Choi, Junesang"https://zbmath.org/authors/?q=ai:choi.junesangSummary: Recently \textit{H. Alzer} and \textit{J. Choi} [J. Math. Anal. Appl. 484, No. 1, Article ID 123661, 22 p. (2020; Zbl 1437.11123)] proposed and studied a set of the four Euler sums being linear with parameters. These sums are parametric extensions of \textit{P. Flajolet} and \textit{B. Salvy}'s four kinds of Euler sums being linear [Exp. Math. 7, No. 1, 15--35 (1998; Zbl 0920.11061)]. The purpose of this paper is to extend the set of the four Euler sums being linear with parameters. Then, we look at several characteristics of the set of the four extended Euler sums being linear with parameters, including reducibility, series involving the zeta functions, and other expressions for their specific instances. We discover here that two well-known and long-established topics, Euler sums and series involving the zeta functions, exhibit specific relationships.Log-hyperbolic tangent integrals and Euler sumshttps://zbmath.org/1491.110812022-09-13T20:28:31.338867Z"Sofo, Anthony"https://zbmath.org/authors/?q=ai:sofo.anthonySummary: An investigation into the representation of integrals involving the product of the logarithm and the \(\tanh^{-1}\) functions will be undertaken in this paper. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed.Consecutive primes in short intervalshttps://zbmath.org/1491.110822022-09-13T20:28:31.338867Z"Radomskii, Artyom O."https://zbmath.org/authors/?q=ai:radomskii.artyom-oLet \(p_i\) be the \(i\)-th prime. In this paper, the author proves the following result:
Theorem. There exist two absolute positive constants \(c\) and \(C\) such that for any \(0<\varepsilon <1\), if \(x,y\) are real numbers and \(a,q\) are coprime integers with \[c_0(\varepsilon)\le y\le \log x,\quad 1\le m\le c\varepsilon \log y, \quad 1\le q\le y^{1-\varepsilon },\] then the number of \(p_n\) with \(\frac x2 <p_n\le x\) such that \(p_{n+m}-p_n\le y\) and \( p_n\equiv \cdots \equiv p_{n+m}\equiv a\pmod q\) does not exceed \[\pi (x) \left( \frac{y}{2q\log x} \right)^{\exp (Cm)},\] where \(c_0(\varepsilon)\) is a positive constant depending only on \(\varepsilon \).
This extends an earlier result of \textit{J. Maynard} [Compos. Math. 152, No. 7, 1517--1554 (2016; Zbl 1382.11074)]. The author also gives the following corollary.
Corollary. There is a positive constant \(C\) such that if \(m\) is a positive integer and \(x,y\) are two real numbers with \(\exp (Cm) \le y\le \log x \), then \[\# \left\{ \frac x2 <p_n\le x : p_{n+m}-p_n\le y\right\} \ge \pi (x) \left( \frac{y}{2q\log x} \right)^{\exp (Cm)}.\]
This paper contains the considerable details of related basic notations and results. The proof of the above theorem is an application of Maynard's Proposition 6.1 [loc. cit.].
Reviewer: Yong-Gao Chen (Nanjing)Coordinate distribution of Gaussian primeshttps://zbmath.org/1491.110832022-09-13T20:28:31.338867Z"Friedlander, John"https://zbmath.org/authors/?q=ai:friedlander.john-b"Iwaniec, Henryk"https://zbmath.org/authors/?q=ai:iwaniec.henrykOne is interested in how many primes \(\pi = a +2bi\) in the ring \(\mathbb{Z}[i]\) have their coordinates restricted to special integers. Here one investigates the case when \(a\) is prime and \(b\) is almost-prime, namely one studies \[ G_r(x) = \sum_{4k^2 + l^2 \leq x} \Lambda_r(k) \Lambda(l) \Lambda(4k^2 + l^2), \] where \(r \geq 1\) is natural and \(\Lambda\) and \(\Lambda_r\) denote the von Mangoldt and generalized von Mangoldt function respectively. With the technology developed in this paper one is able to adequately estimate \(G_r(x)\) when \(r \geq 7\); one has \(G_7(x) \asymp x(\log x)^6\).
The starting ingredient of the proof is the sieve inequality Proposition 6.1 allowing one to estimate a modified version of the sum for \(G_r\) by a sum of congruence sums and a bilinear form. The analysis of the latter two quantities then makes up most of the paper.
Reviewer: Gregory Debruyne (Champaign)Multiplicative functions in large arithmetic progressions and applicationshttps://zbmath.org/1491.110842022-09-13T20:28:31.338867Z"Fouvry, Étienne"https://zbmath.org/authors/?q=ai:fouvry.etienne"Tenenbaum, Gérald"https://zbmath.org/authors/?q=ai:tenenbaum.geraldAuthors' abstract: We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by \textit{S. Drappeau} and \textit{B. Topacogullari} [Algebra Number Theory 13, No. 10, 2383--2425 (2019; Zbl 1472.11252)] for arithmetical correlations; a theorem of Erdős-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by \(\tau (n-1)\) where \(\tau\) denotes the divisor function.
Reviewer: Yong-Gao Chen (Nanjing)Asymptotic bounds for factorizations into distinct partshttps://zbmath.org/1491.110852022-09-13T20:28:31.338867Z"Lebowitz-Lockard, Noah"https://zbmath.org/authors/?q=ai:lebowitz-lockard.noahLet \(f(n)\) be the number of unordered factorizations of an integer \(n\) into parts greater than \(1\) and let \(F(n)\) be the number of unordered factorizations of \(n\) into distinct parts greater than \(1\). In this paper, the author obtains some lower and upper bounds for \(f(n)\) and \(F(n)\). Meanwhile, he proves that \(f(n)\leq\sqrt{n}F(n)\) for any positive integer \(n\). The author also obtains an upper bound for \(G(n)\), the number of ordered factorizations of \(n\) into distinct parts greater than \(1\).
Reviewer: Mehdi Hassani (Zanjan)Generalized divisor functions in arithmetic progressions: I.https://zbmath.org/1491.110862022-09-13T20:28:31.338867Z"Nguyen, David T."https://zbmath.org/authors/?q=ai:nguyen.david-tLet \(n\geqslant1\), \(k\geqslant 1\) be integers, and let \(\tau_k\) denote the \(k\)-fold divisor function, i.e. \[ \tau_k(n)=\sum_{n_1n_2\cdots n_k=\,n}1, \] where the sum runs over ordered \(k\)-tuples \((n_1, n_2,\dots, n_k)\) of positive integers for which \(n_1n_2\cdots n_k = n\).
The author of the paper under review proves the following theorem on distribution of function \(\tau_k\).
Let \(w=1/1168\), \(\theta_k=\min\{1/(12(k+2)),w^2\}\) and \[ \mathcal{D}=\bigg\{d\geqslant 1, (d,a)=1, |\mu(d)|=1, \Big(d,\hspace{-2mm} \prod_{p\,<\,X^{w^2}}\Big)<X^w, \Big(d,\hspace{-1mm}\prod_{\ p\,<\,X^{w}}\Big)>X^{71/584}\bigg\} \] where \(a\neq 0\) and \(\mu\) is the Miobius function. Then for each \(k\geqslant 4\) it holds that \[ \sum_{\substack{d\in\mathcal{D}\\ d<X^{293/584}}}\bigg{|}\sum_{\substack{n\leqslant X \\ n\equiv a\bmod d}}\tau_k(n)-\frac{1}{\varphi(d)}\sum_{\substack{n\leqslant X \\ (n,d)=1}}\tau_k(n)\bigg{|}\ll X^{1-\theta_k}, \] where \(\varphi\) is the Euler's totient function, and the implied constant is effective, and depends at most on \(a\) and \(k\).
In addition, the author presents a number of conclusions from the main result and examines the influence of the \textit{Generalized Riemann Hypothesis} and the \textit{Generalized Lindelöf Hypothesis} on the main result.
Reviewer: Jonas Šiaulys (Vilnius)Goldbach-Linnik type problems with unequal powers of primeshttps://zbmath.org/1491.110872022-09-13T20:28:31.338867Z"Zhu, Li"https://zbmath.org/authors/?q=ai:zhu.liThe main result of the paper improves from \(22\) to \(17\) the number \(N\) of powers of \(2\) that are required to represent a large even integer \(n\) as a sum of \(2\) squares, \(2\) cubes, \(2\) fourth powers, and \(N\) powers of \(2\).
Reviewer: Luis Gallardo (Brest)Visible lattice points along curveshttps://zbmath.org/1491.110882022-09-13T20:28:31.338867Z"Liu, Kui"https://zbmath.org/authors/?q=ai:liu.kui"Meng, Xianchang"https://zbmath.org/authors/?q=ai:meng.xianchangA lattice point \((m,n) \in \mathbb{N}\times \mathbb{N}\) is said to be visible from the lattice point \((u,v) \in \mathbb{N}\times \mathbb{N}\), if there do not exist any other integer lattice points on the straight line segment joining \((m,n)\) and \((u,v)\). A historic result due to
\textit{J. J. Sylvester} [C. R. Acad. Sci., Paris 96, 409--413 (1883; JFM 15.0132.01)] says that the proportion of lattice points that are visible from the origin \((0,0)\) is \(1/\zeta(2)=6/\pi^{2}\).
In this present work the authors consider the distribution of lattice points which are visible from multiple base points simultaneously. They employ the definition that for any positive integer \(k\) and integer lattice points \((u,v)\) , \((m,n) \in \mathbb{N}\times \mathbb{N}\), where \(r\in \mathbb{Q}\) is given by \(n-v=r(m-u)^{k}\) and \(C\) be the curve \(y-v=r(x-u)^{k}\), then if there are no integer lattice points lying on the segment of \(C\) between points \((m,n)\) and \((u,v)\), then \((m,n)\) is defined to be Level-1 \(k\)-visible to \((u,v)\) . Furthermore, if there is at most one integer lattice point lying on the segment of \(C\) between points \((m,n)\) and \((u,v)\), then \((m,n)\) is said to be Level-2 \(k\)-visible to \((u,v)\) .
It follows that if a point \((m,n)\) is Level-1 or Level-2 \(k\)-visible to the point \((u,v)\) along the curve \(y-v=r(x-u)^{k}\), then \((u,v)\) is also Level-1 or Level-2 \(k\)-visible to \((m,n)\) , respectively, along the curve \(y-n= (-1)^{k+1}r(x-m)^{k}.\)
Let \(S\) be a given set of integer lattice points in the plane. The authors generalise the above definitions, and say that an integer lattice point \((m,n)\) is Level-1 \(k\)-visible to \(S\) if it belongs to the set
\(V_{k}^{1}(S) :=\) \{\((m\) , \(n)\in \mathbb{N}\times \mathbb{N}\) : \((m , n)\) is \(k\)-visible to every point in \(S\) \}.
Similarly, a point \((m,n) \in \mathbb{N}\times \mathbb{N}\) is defined to be Level-2 \(k\)-visible to \(S\) if it belongs to the set
\(V_{k}^{2}(S) :=\) \{\((m\) , \(n)\in \mathbb{N}\times \mathbb{N}\) : \((m,n)\) is Level-2 \(k\)-visible to every point in \(S\) \}.
For \(x\geq 2\), the authors consider visible lattice points along curves in the square \([\)1, \(x]\times[1,x],\) with the notation
\(N_{k}^{1}(S,x):=\#\{(m,n)\in V_{k}^{1}(S):m,n\leq x\},\)
and
\(N_{k}^{2}(S,x):=\#\{(m,n)\in V_{k}^{2}(S):m,n\leq x\}.\)
They focus on the interesting case when the points of \(S\) are pairwise \(k\)-visible to each other, so that the cardinality of \(S\) can't be too large and is bounded by \(\# S\leq 2^{k+1}\). Their main results (Theorems 1.1 and 1.2) give asymptotic formulas for \(N_{k}^{1}(S,x)\) and \(N_{k}^{2}(S,x)\).
Reviewer: Matthew C. Lettington (Cardiff)On additive and multiplicative decompositions of sets of integers composed from a given set of primes, I. (additive decompositions)https://zbmath.org/1491.110892022-09-13T20:28:31.338867Z"Győry, K."https://zbmath.org/authors/?q=ai:gyory.kalman"Hajdu, L."https://zbmath.org/authors/?q=ai:hajdu.lajos"Sárközy, A."https://zbmath.org/authors/?q=ai:sarkozy.andrasWe introduce the terminology used by the authors. A finite or infinite set \(\mathcal{A}\) of non-negative integers is called a(dditively)-reducible if there are subsets \(\mathcal{B}\), \(\mathcal{C}\) of \(\mathcal{A}\) of cardinality at least \(2\) such that \(\mathcal{A}=\mathcal{B}+\mathcal{C}:=\{ b+c :\, b\in\mathcal{B},c\in\mathcal{C}\}\). Then \(\mathcal{B}+\mathcal{C}\) is called an a-decomposition of \(\mathcal{A}\). If such sets \(\mathcal{B}\), \(\mathcal{C}\) do not exist, the set \(\mathcal{A}\) is called \(a\)-primitive. If such sets \(\mathcal{B}\), \(\mathcal{C}\) with at least one of them finite do not exist, the set \(\mathcal{A}\) is called a-F-primitive. lastly, the set \(\mathcal{A}\) is called totally \(a\)-primitive, resp. totally a-F-primitive, if every set \(\widetilde{\mathcal{A}}\) that differs from \(\mathcal{A}\) by at most finitely many elements is \(a\)-primitive, resp. a-F-primitive.
Given a set of primes \(\mathcal{P}\) (finite or infinite), denote by \(\mathcal{R}(\mathcal{P})\) the set of positive integers not divisible by any prime outside \(\mathcal{P}\). \textit{C. Elsholtz} and \textit{A. J. Harper} [Trans. Am. Math. Soc. 367, No. 10, 7403--7427 (2015; Zbl 1325.11098)] proved that if \(\mathcal{P}\) is finite, then \(\mathcal{R}(\mathcal{P})\) is totally a-primitive. Further, they showed that there exist infinite sets of primes \(\mathcal{P}\) such that \(\mathcal{R}(\mathcal{P})\) is totally a-primitive. The authors make this more precise as follows:
Theorem 1. Let \(\mathcal{P}\) be a non-empty set of primes with the property that there exists \(x_0\) such that \(\#\{ p\in\mathcal{P}:\, p\leq x\}<\frac{1}{51}\log\log x\) for \(x>x_0\). Then \(\mathcal{R}(\mathcal{P})\) is totally a-primitive.
The proof uses Beukers' and Schlickewei's uniform upper bound \(2^{16r+8}\) for the number of solutions of the equation \(x+y=1\) with \(x,y\) belonging to a subgroup of \(\mathbb{Q}^*\) of rank \(r\) [\textit{F. Beukers} and \textit{H. P. Schlickewei}, Acta Arith. 78, No. 2, 189--199 (1996; Zbl 0880.11034)].
The authors also investigated what is happening if \(\mathcal{P}\) is a much denser set of primes, e.g., when \(\mathcal{P}=\mathcal{Q}^c\) is the complement in the set of primes of a small set of primes \(\mathcal{Q}\). By more elementary arguments they prove the following:
Theorem 2. Let \(\mathcal{Q}\) be any finite set of primes and \(\mathcal{P}=\mathcal{Q}^c\). Let \(t\) be either an integer \(\geq 2\) or \(\infty\). Then \(\mathcal{R}(\mathcal{P})\) has an a-decomposition \(\mathcal{A}+\mathcal{B}\), where \(\mathcal{A}\) is infinite, and \(\mathcal{B}\) has cardinality \(t\).
Theorem 3. Let \(f:\,\mathbb{Z}_{>0}\to\mathbb{R}\) be any non-decreasing function with \(\lim_{n\to\infty} f(n)=\infty\). Then there exists an infinite set of primes \(\mathcal{Q}\) with \(\#\{ p\in\mathcal{Q}:\, p\leq n\}<f(n)\) for all \(n\) such that \(\mathcal{P}=\mathcal{Q}^c\) is infinite, and \(\mathcal{R}(\mathcal{P})\) is totally a-F-primitive.
Reviewer: Jan-Hendrik Evertse (Leiden)Linked partition ideals and the Alladi-Schur theoremhttps://zbmath.org/1491.110902022-09-13T20:28:31.338867Z"Andrews, George E."https://zbmath.org/authors/?q=ai:andrews.george-eyre"Chern, Shane"https://zbmath.org/authors/?q=ai:chern.shane"Li, Zhitai"https://zbmath.org/authors/?q=ai:li.zhitaiLet \(D(n)\) denote the number of partitions of the positive integer \(n\) of the form \(\mu_1+\mu_2+\ldots+\mu_s\), where \(\mu_i-\mu_{i+1}\ge 3\) with strict inequality if \(3\mid\mu_i\) and let \(C(n)\) denote the number of partitions of \(n\) into odd parts with none appearing more than twice. The Alladi-Schur theorem asserts that, for all \(n\), \(D(n)=C(n)\). We also recall that an Andrews-Gordon series is of the form \(\sum_{n_1,\ldots,n_r\ge 0}\frac{(-1)^{L_1(n_1,\ldots,n_r)}q^{Q(n_1,\ldots,n_r)+L_2(n_1,\ldots,n_r)}}{(q^{A_1};q^{A_1})_{n_1}\ldots(q^{A_r};q^{A_r};q^{A_r})_{n_r}}\), where \(L_1\) and \(L_2\) are linear forms and \(Q\) is a quadratic form in \(n_1,\ldots,n_r\) and the \(q\)-Pochhammer symbol is defined for \(n\in\mathbb{N}\cup\{\infty\}\) by \((A;q)_n=\prod_{k=0}^{n-1}(1-Aq^k)\).
Authors' abstract: ``Let \(\mathcal{L}\) denote the set of integer partitions that differ by at least \(3\), with the added constraint that no two consecutive multiples of \(3\) occur as parts. We derive trivariate generating functions of Andrews-Gordon type for partitions in \(\mathcal{L}\) with both the number of parts and the number of even parts counted. In particular, we provide an analytic counterpart of Andrews' recent refinement of the Alladi-Schur theorem.''
Reviewer: Ljuben Mutafchiev (Sofia)On the existence of Graham partitions with congruence conditionshttps://zbmath.org/1491.110912022-09-13T20:28:31.338867Z"Kim, Byungchan"https://zbmath.org/authors/?q=ai:kim.byungchan"Kim, Ji Young"https://zbmath.org/authors/?q=ai:kim.ji-young"Lee, Chong Gyu"https://zbmath.org/authors/?q=ai:lee.chong-gyu"Lee, Sang June"https://zbmath.org/authors/?q=ai:lee.sangjune"Park, Poo-Sung"https://zbmath.org/authors/?q=ai:park.poo-sung\textit{Ronald L. Graham} [J. Aust. Math. Soc. 3, 435--441 (1963; Zbl 0142.01304)] was the first to study the partition with a specific condition on the reciprocal sum of the parts. The authors call a partition a Graham partition if the reciprocal sum of its parts is \(1\). In this paper, the authors investigate Graham partitions into parts satisfying some congruence conditions.
Reviewer: Mircea Merca (Cornu de Jos)On the \(k\)th root partition functionhttps://zbmath.org/1491.110922022-09-13T20:28:31.338867Z"Li, Ya-Li"https://zbmath.org/authors/?q=ai:li.yali"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.1For any positive integer \(k\), the authors consider \(p_{1/k}(n)\) to be the number of partitions of the integer \(n\) into \(k\)th roots, i.e., the number of solutions to the equation \[n=\lfloor \sqrt[k]{a_1} \rfloor + \lfloor \sqrt[k]{a_2} \rfloor +\cdots + \lfloor \sqrt[k]{a_d} \rfloor,\] with \(d\geq 1\) and \(a_1\geq a_2\geq \cdots \geq a_d \geq 1\).
Inspired by \textit{G. Tenenbaum} et al. [J. Number Theory 204, 435--445 (2019; Zbl 1416.05034)], the authors establish an asymptotical development of \(p_{1/k}(n)\) by the saddle-point method. In this context, the authors prove that \(p_{1/k}(n)\) is odd for infinitely many integers \(n\) and \(p_{1/k}(n)\) is even for infinitely many integers \(n\). \textit{O. Kolberg}'s result [Math. Scand. 7, 377--378 (1960; Zbl 0091.04402)] on the parity on Euler's partition function \(p(n)\) is the case \(k=1\) of this result.
Reviewer: Mircea Merca (Cornu de Jos)The asymptotic number of weighted partitions with a given number of partshttps://zbmath.org/1491.110932022-09-13T20:28:31.338867Z"Stark, Dudley"https://zbmath.org/authors/?q=ai:stark.dudleyIn combinatorics, a weighted partition \(\lambda_b\) of the positive integer \(n\) is described as follows: for a given sequence of non-negative intergers \((b_k)_{k\ge 1}\), a part of size \(k\) appears in exactly \(b_k\) possible types (colors) in the partition \(\lambda_b\). It turns out that in several asymptotic enumeration problems concerning weighted partitions, one can drop the assumption that \(b_k\ge 0\) are integers.
Author's abstract: For a given sequence \(b_k\) of non-negative real numbers, the number of weighted partitions of the positive integer \(n\) having \(m\) parts \(c_{n,m}\) has bivariate generating function equal to \(\prod_{k=1}^\infty(1-yz^k)^{-b_k}\). Under the assumption that \(b_k\sim Ck^{r-1}, r>0\), and related conditions on the Dirichlet generating function of the weights \(b_k\), we find asymptotics of \(c_{n,m}\) when \(m=m(n)\) satisfies \(m=o(n^{r-1})\) and \(\lim_{n\to\infty}m/\log^{3+\epsilon}{n}=\infty, \varepsilon>0\).
Reviewer: Ljuben Mutafchiev (Sofia)Proofs of some conjectures on the reciprocals of the Ramanujan-Gordon identitieshttps://zbmath.org/1491.110942022-09-13T20:28:31.338867Z"Bian, Min"https://zbmath.org/authors/?q=ai:bian.min"Tang, Dazhao"https://zbmath.org/authors/?q=ai:tang.dazhao"Xia, Ernest X. W."https://zbmath.org/authors/?q=ai:xia.ernest-x-w"Xue, Fanggang"https://zbmath.org/authors/?q=ai:xue.fanggangRamanujan and Gordon independently discovered the following two nice product-to-sum identities:
\[
\sum_{n=-\infty}^{\infty}(6n+1)q^{n(3n+1)/2}=\frac{(q;q)_{\infty}^5}{(q^2;q^2)_{\infty}^2},
\]
\[
\sum_{n=-\infty}^{\infty}(3n+1)q^{n(3n+2)}=\frac{(q;q)_{\infty}^2(q^4;q^4)_{\infty}^2}{(q^2;q^2)_{\infty}},
\]
which are called the Ramanujan-Gordon identities. Define two partition functions \(RG_1(n)\) and \(RG_2(n)\) by
\[
\sum_{n=0}^{\infty}RG_1(n)q^n=\frac{(q^2;q^2)_{\infty}^2}{(q;q)_{\infty}^5},
\]
\[
\sum_{n=0}^{\infty}RG_2(n)q^n=\frac{(q^2;q^2)_{\infty}}{(q;q)_{\infty}^2(q^4;q^4)_{\infty}^2},
\]
whose generating functions are the reciprocals of \(\sum_{n=-\infty}^{\infty}(6n+1)q^{n(3n+1)/2}\) and \(\sum_{n=-\infty}^{\infty}(3n+1)q^{n(3n+2)}\) respectively.
In the paper under review, via Ramanujan's modular equation of fifth degree, the authors prove the following four congruences modulo 25 for \(RG_1(n)\) and \(RG_2(n)\): \begin{eqnarray*} RG_1(125n+74)\equiv 0\ (\mathrm{mod\ }25), \\
RG_1(125n+124)\equiv 0\ (\mathrm{mod\ }25), \\
RG_2(125n+92)\equiv 0\ (\mathrm{mod\ }25),\\
RG_2(125n+117)\equiv 0\ (\mathrm{mod\ }25), \end{eqnarray*} which were conjectured by \textit{B. L. Lin} and \textit{A. Y. Wang} [Colloq. Math. 154, No. 1, 137--148 (2018; Zbl 1429.11190)]. In addition, the authors also establish some new congruences modulo 25 for \(RG_1(n)\) and \(RG_2(n)\). Their proof is based on Newman's identities. For example, they deduce that for any integer \(n\geq 0\),
\[
RG_1\left(\frac{23375n(3n+1)}{2}+974\right)\equiv RG_1\left(\frac{23375n(3n+5)}{2}+24349\right)\equiv 0\ (\mathrm{mod\ }25).
\]
\[
RG_2(2000n(3n+1)+167)\equiv RG_2(2000n(3n+5)+4167)\equiv 0\ (\mathrm{mod\ }25).
\]
Reviewer: Donna Q. J. Dou (Changchun)Weighted partition rank and Crank moments. III. A list of Andrews-Beck type congruences modulo \(5, 7, 11\) and \(13\)https://zbmath.org/1491.110952022-09-13T20:28:31.338867Z"Chern, Shane"https://zbmath.org/authors/?q=ai:chern.shaneIn order to combinatorially explain the famous Ramanujan congruences
\[
p(5n+4)\equiv 0 \mod 5, \quad p(7n+5)\equiv 0 \mod 7, \ \text{ and } \ p(11n+6)\equiv 0 \mod 11,
\]
for the number \(p(n)\) of the partitions of \(n\), \textit{F. J. Dyson} [Eureka 8, 10--15 (1944)] defined the rank of a partition as the largest part minus the number of parts. He conjectured that this statistic could be utilized to show the mod 5 and mod 7 congruences, which was later confirmed by \textit{A. O. L. Atkin} and \textit{P. Swinnerton-Dyer} [Proc. Lond. Math. Soc. (3) 4, 84--106 (1954; Zbl 0055.03805)]. Dyson also observed that the rank did not separate the partitions of \(11n+6\) into 11 equal classes and he conjectured the existence of a crank for the mod 11 congruence. \textit{G. E. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167--171 (1988; Zbl 0646.10008)] defined the crank of a partition as the largest part if the partition contains no ones and otherwise the number of parts larger than the number of ones minus the number of ones.
Let \(NT(r, k, n)\) denote the total number of parts in the partitions of \(n\) with rank congruent to \(r\) modulo \(k\) and let \(M_{\omega}(r, k, n)\) count the total appearances of ones in the partitions of \(n\) with crank congruent to \(r\) modulo \(k\). For \(NT(r, k, n)\), \textit{G. E. Andrews} [Int. J. Number Theory 17, No. 2, 239--249 (2021; Zbl 1465.11200)] proved surprising congruences (conjectured by George Beck). \textit{S. H. Chan}, \textit{R. Mao}, and \textit{R. Osburn} [J. Math. Anal. Appl. 495, No. 2, Article ID 124771, 14 p. (2021; Zbl 1464.11106)] conjectured many congruences and relations for \(NT(r, k, n)\) and \(M_{\omega}(r, k, n)\).
In the paper under review the author provides a list of over 70 congruences for \(NT(r, k, n)\) and \(M_{\omega}(r, k, n)\) modulo 5, 7, 11, and 13. E.g., the author proves that
\[
\begin{multlined}
NT(3, 13, 13n+2) - NT(10, 13, 13n+2) + 9 NT(4, 13, 13n+2)\\ - 9 NT(9, 13, 13n+2)+ 6 NT(5, 13, 13n+2) - 6 NT(8, 13, 13n+2) \\ + 8 NT(6, 13, 13n+2) - 8 NT(7, 13, 13n+2) \equiv 0 \mod 13
\end{multlined}
\]
and
\[
\sum_{r=1}^6 r \big( M_{\omega}(r, 13, 13n) - M_{\omega}(13-r, 13, 13n)\big) \equiv 0 \mod 13.
\]
Reviewer: Mihály Szalay (Budapest)Parity of the coefficients of certain eta-quotientshttps://zbmath.org/1491.110962022-09-13T20:28:31.338867Z"Keith, William J."https://zbmath.org/authors/?q=ai:keith.william-j"Zanello, Fabrizio"https://zbmath.org/authors/?q=ai:zanello.fabrizioThis is a fascinating and beautifully written paper on some very classical (and still very mysterious!) aspects of modular forms. For integers \(j\geq 1\), let \(f_j := (1-q^j)(1-q^{2j})(1-q^{3j})\cdots \in \mathbb{Z}[[q]]\). Consider ratios \(F(q)\) of the form \((f_{a_1}\cdots f_{a_m})/(f_{b_1}\cdots f_{b_n})\), with \(a_1,\dots,a_m\) and \(b_1,\dots,b_n\) being arbitrary sequences of positive integers. Say \(F(q)\) has odd density \(\delta\) if its odd coefficients occur with density exactly \(\delta\). Say \(F\bmod{2}\) is lacunary if it \(F\) has odd density \(0\).
[\textit{T. R. Parkin} and \textit{D. Shanks}, Math. Comput. 21, 466--480 (1967; Zbl 0149.28501)] conjectured that \(1/f_1\), the generating series of the partition function, has odd density \(1/2\). The authors make a vast novel generalization (Conjecture~4) to all ratios \(F(q)\) above, conjecturing that (i) the odd density \(\delta_F\) always exists, and satisfies \(\delta_F\leq 1/2\); (ii) if \(\delta_F = 1/2\), then the coefficients of \(F\) must have odd density \(1/2\) when restricted to any nontrivial polynomial sequence (e.g.,~any non-constant arithmetic progression); and (iii) if \(\delta_F < 1/2\) (e.g.,~if \(F\bmod{2}\) is lacunary), then the coefficients of \(F\bmod{2}\) must vanish identically along some non-constant arithmetic progression.
The conjecture highlights the significance of even progressions (i.e.,~non-constant arithmetic progressions along which the coefficients of \(F\bmod{2}\) vanish). It is thus of interest to try to find even progressions for various ratios \(F\), or to conjecture their nonexistence otherwise. This, among other things, is what the present authors, together with previous literature, do for the generating series \(B_m(q) := f_m/f_1\) for \(m\)-regular partitions (partitions with no parts divisible by \(m\)), for positive integers \(m\leq 28\). We refer the reader to p.~278 (especially the table within), and to \S2, for a description of the authors' very substantial contributions to this task; the authors prove (and give many further conjectures on) many results on lacunarity, self-similarity, and congruences.
The tools used include a wide variety of identities (e.g.,~the following: Euler's Pentagonal Number Theorem, for \(f_1\bmod{2}\); the Frobenius congruence \(f_j^2\equiv f_{2j}\bmod{2}\); expansions, from [\textit{N. Robbins}, Fibonacci Q. 38, No. 1, 39--48 (2000; Zbl 0947.11030)] and [\textit{M. D. Hirschhorn}, Ramanujan J. 4, No. 2, 129--135 (2000; Zbl 0988.11049)], of the \(t\)-core series \(f_t^t/f_1\bmod{2}\); and certain dissection identities from [\textit{S. D. Judge} et al., Ann. Comb. 22, No. 3, 583--600 (2018; Zbl 1440.11195)]); auxiliary modular forms and Hecke operators; the Sturm bound for congruences between modular forms; and the lacunarity of values of positive-definite binary quadratic forms (which, in a quantitative form, goes back to Landau).
Finally, it is worth quoting the following part of the abstract: ``All of our work is consistent with a new, overarching conjecture that we present for arbitrary eta-quotients, greatly extending Parkin-Shanks' classical conjecture for the partition function. We pose several other open questions throughout the paper, and conclude by suggesting a list of specific research directions for future investigations in this area.''
Reviewer: Victor Wang (Princeton)Identities, inequalities and congruences for odd ranks and \(k\)-marked odd Durfee symbolshttps://zbmath.org/1491.110972022-09-13T20:28:31.338867Z"Xia, Ernest X. W."https://zbmath.org/authors/?q=ai:xia.ernest-x-w\textit{G. E. Andrews} [Invent. Math. 169, No. 1, 37--73 (2007; Zbl 1214.11116)] introduced the odd rank of odd Durfee symbols. Let \(N^0(m, n)\) denote the number of odd Durfee symbols of \(n\) with odd rank \(m\), and \(N^0(r, m; n)\) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, the generating functions is given of odd Durfee symbols of n with odd rank congruent to r modulo m for \(m=12\) by utilizing some identities involving Appell-Lerch sums and a universal mock theta function. Based on these formulas, they determine the signs of \(N^0(r, 12; 4n + t) -N^0(s, 12; 4n + t)\) for all \(0\leq r, s\leq 6\) and \(0\leq t\leq 3\).
Reviewer: Rózsa Horváth-Bokor (Budakalász)On power integral bases for certain pure number fields defined by \(x^{2\cdot 3^k}-m\)https://zbmath.org/1491.110982022-09-13T20:28:31.338867Z"El Fadil, Lhoussain"https://zbmath.org/authors/?q=ai:el-fadil.lhoussainAn algebraic number field $K$ is said to be monogenic if its ring of integers $\mathbb{Z}_{K}$ is equal to the subring $\mathbb{Z}[\alpha]$ of $K$ generated by some element $\alpha\in K$.
The problem of testing the monogeneity of number fields has been intensively studied for a long time, and in the paper under review the author continues to investigate such a question for a pure number field $\mathbb{Q}(\theta)$ of degree $2\cdot 3^{k}$, where $k\in\mathbb{N}$, and $\theta$ is a root of a polynomial of the form $x^{2\cdot 3^{k}}-m$ with$\ m\neq\pm 1$ a square free integer. More precisely, he proves that if $m\equiv 2,3\pmod 4$ and $m\neq\pm 1\pmod 9$, then $\mathbb{Z}_{\mathbb{Q}(\theta)}=\mathbb{Z}[\theta]$, while if $m\equiv 1\pmod 4$ or $m\equiv 1\pmod 9$ or $(m\equiv-1\pmod{81}$ and $k\geq 3)$, then $\mathbb{Q}(\theta)$ is not monogenic. Also, he shows that $\mathbb{Q}(\theta)$ is monogenic if and only if $m\equiv 2,3\pmod 4$ and $m\neq 1\pmod 9$, whenever $k<3$.
These results, based on the Newton's polygon method, are analogous to those obtained by the author himself in a series of articles and extend some of them.
Reviewer: Toufik Zaïmi (Riyadh)On the discriminant of pure number fieldshttps://zbmath.org/1491.110992022-09-13T20:28:31.338867Z"Jakhar, Anuj"https://zbmath.org/authors/?q=ai:jakhar.anuj"Khanduja, Sudesh K."https://zbmath.org/authors/?q=ai:khanduja.sudesh-k|khanduja.sudesh-kaur"Sangwan, Neeraj"https://zbmath.org/authors/?q=ai:sangwan.neerajThe article is concerned with the determination of the discriminant of a pure number field \(K=\mathbb{Q}(\sqrt[3]{-2}n{a})\), providing the most general result up to the present moment. The authors assume the only restriction that for every prime \(p\) dividing \(n\), either \(p\) does not divide \(a\) or \(v_p(a)\) is coprime to \(p\).
Given the prime factorizations \(n=p_1^{s_1}\cdots p_k^{s_k}\), \(\left|a\right|=q_1^{t_1}\cdots q_l^{t_l}\), let \(m_j=\gcd(n,t_j)\), \(n_i=n/p_i^{s_i}\), \(r_i=v_{p_i}(a^{p_i-1}-1)-1\). The discriminant \(d_K\) of \(K\) is, under the staten restriction: \[ d_k=(-1)^{(n-1)(n-2)/2}\operatorname{sgn}(a^{n-1}) \left(\prod_{i=1}^{k}p_i^{v_i}\right) \left(\prod_{j=1}^{l}q_j^{n.m_j}\right), \] where \(v_i=ns_i\) if \(r_i=0\) and \(v_i=n s_i-\sum_{j=1}^{\min{r_i,s_i}}p_i^{s_i-j}\) for \(r_i> 0\). As a corollary, the authors show that the powers of \(\sqrt[3]{-2}n{a}\) give an integral basis of \(K\) if and only if \(a\) is squarefree and all the \(r_i=0\). These results were already obtained by \textit{T. Alden Gassert} [Albanian J. Math. 11, 3--12 (2017; Zbl 1392.11082)] using the higher order Newton polygons of Montes, but in the present article the authors use only the classical results of Ore on Newton polygons.
Reviewer: Jordi Guárdia (Vilanova i la Geltrú)Cubic function fields with prescribed ramificationhttps://zbmath.org/1491.111002022-09-13T20:28:31.338867Z"Karemaker, Valentijn"https://zbmath.org/authors/?q=ai:karemaker.valentijn"Marques, Sophie"https://zbmath.org/authors/?q=ai:marques.sophie"Sijsling, Jeroen"https://zbmath.org/authors/?q=ai:sijsling.jeroenLet \(K\) be a function field with a perfect field \(k\) as its field of constants and let \(L/K\) be a separable geometric extension of degree \(3\). It is assumed that the characteristic of \(K\) is different from \(3\). Usually the construction or the study of these extensions are by means of the discriminant \(\delta\) of \(L/K\). In the paper under review, the authors classify extensions with a given set of ramified places instead of those with a given discriminant. The extension \(L/K\) is called {\em purely cubic} if there exists a generator \(y\) whose minimal polynomial over \(K\) is of the form \(X^3-\beta\), otherwise it is called {\em impurely cubic}. When \(L/K\) is impurely cubic, \(L/K\) admits a purely cubic closure \(K'\), which is a degree two extension of \(K\) such that \(LK'/K'\) is purely cubic.
The main result is Theorem 4.1, which establishes that if \(K=k(x)\) is a rational function field and if \(K'\) is of genus zero and \(T\) is a given set of places of \(K'\), then there exists a cubic extension \(L/K\) with purely cubic closure \(K'\) and triple ramification precisely at the places in \(T\) if and only if all places in \(T\) split in \(K'\).
The idea is to determine an extension \(L/K\) by descending from its base change \(LK'/K'\) to the purely cubic closure \(K'\).
In Section 6 the authors consider the case where \(K\) is not of genus \(0\) by constructing Parshin covers and applying the methods used for the case \(K=k(x)\). For cubic function fields of genus at most one, are described the twists and isomorphism classes obtained when Möbius transformations on \(K\) are allowed.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)Normal elements in the Iwasawa algebras of Chevalley groupshttps://zbmath.org/1491.111012022-09-13T20:28:31.338867Z"Han, Dong"https://zbmath.org/authors/?q=ai:han.dong"Ray, Jishnu"https://zbmath.org/authors/?q=ai:ray.jishnu"Wei, Feng"https://zbmath.org/authors/?q=ai:wei.fengLet \(p\) be an odd prime and let \(G\) be a semisimple, simply connected, split Chevalley group over \(\mathbb Z_p\) that is of one of the following types: \(A_{\ell}\) with \(\ell\ge 1\), \(B_{\ell}\) with \(\ell\ge 2\), \(C_{\ell}\) with \(\ell\ge 2\), \(D_{\ell}\) with \(\ell\ge 3\), \(E_6\), \(E_7\), \(E_8\), \(F_4\), \(G_2\). Let \(G(1)\) be the kernel of the reduction of \(G(\mathbb Z_p)\) mod \(p\) and let \(\Omega_{G(1)}=\lim_{\leftarrow}\mathbb F_p[G(1)/N]\) where \(N\) runs through the open normal subgroups of \(G(1)\). An element \(W\in \Omega_{G(1)}\) is called a normal element if \(W\Omega_{G(1)} = \Omega_{G(1)}W\). The main result of the paper is that \(W\ne 0\) is a normal element if and only if \(W\), as a noncommutative formal power series, has nonzero constant term (and therefore is a unit). The proof relies heavily on work of \textit{M. Lazard} [Publ. Math., Inst. Hautes Étud. Sci. 26, 389--603 (1965; Zbl 0139.02302)] and an explicit presentation of the Iwasawa algebra. Previously. \textit{K. Ardakov} et al. [Adv. Math. 218, No. 3, 865--901 (2008; Zbl 1153.16015); J. Algebra 320, No. 1, 259--275 (2008; Zbl 1162.16013)] used different techniques to prove this result for \(G\) with an extra condition on the prime \(p\).
Reviewer: Lawrence C. Washington (College Park)Galois coinvariants of the unramified Iwasawa modules of multiple \(\mathbb{Z}_p\)-extensionshttps://zbmath.org/1491.111022022-09-13T20:28:31.338867Z"Miura, Takashi"https://zbmath.org/authors/?q=ai:miura.takashi"Murakami, Kazuaki"https://zbmath.org/authors/?q=ai:murakami.kazuaki"Okano, Keiji"https://zbmath.org/authors/?q=ai:okano.keiji"Otsuki, Rei"https://zbmath.org/authors/?q=ai:otsuki.reiLet \(p\) be an odd prime and let \(k\) be a totally real number field. Let \(K\) be a CM-field that is a finite abelian extension of \(k\) of degree prime to \(p\). Let \(K_{\infty}\) be the cyclotomic \(\mathbb Z_p\)-extension of \(K\). The paper is divided into two parts, according to how \(p\) splits in \(K/k\).
In the ``split'' case, it is assumed that \(\mu_p\not\subset K\). Let \(\chi: \text{Gal}(K/k) \to \overline{\mathbb Q}_p^{\times}\) be an odd character and assume there is only one prime ideal \(\mathfrak{p}\) in \(k\) above \(p\) such that \(\chi(\mathfrak{p})=1\). Let \( \widetilde{K}'\) be the maximal multiple \(\mathbb Z_p\)-extension of \(K\) that contains \(K_{\infty}\) and such that \( \widetilde{K}'/K_{\infty}\) is unramified. There exists a subextension \( \widetilde{K}_{\chi}\) of \( \widetilde{K}'/K_{\infty}\) such that \(\text{Gal}( \widetilde{K}_{\chi}/K_{\infty})\) is isomorphic to \(\text{Gal}( \widetilde{K}'/K)^{\chi}\) as \(\text{Gal}(K/k)\)-modules. For a field \(F\), let \(X_F\) denote the Galois group over \(F\) of the maximal unramified abelian \(p\)-extension of \(F\). The authors prove that the order of \((X_{ \widetilde{K}_{\chi}})_{\text{Gal}( \widetilde{K}_{\chi}/K)}\) equals the order of \(\mathcal O_{\chi}/f_{\chi}^*\), where \(f_{\chi}^*\) is the first non-vanishing coefficient of a characteristic power series of the \(\mathcal O_{\chi}[[\text{Gal}(K_{\infty}/K)]]\)-module \((X_{K_{\infty}})^{\chi}\) and \(\mathcal O_{\chi}\) is obtained from \(\mathbb Z_p\) by adjoining the values of \(\chi\).
In the ``non-split'' case, the authors consider an imaginary quadratic field \(K\) in which \(p\) does not split. Let \(A_K\) be the \(p\)-part of the class group of \(K\), let \(L_K\) be the Hilbert \(p\)-class field of \(K\), let \(\lambda\) be the Iwasawa invariant for \(K_{\infty}/K\), and let \( \widetilde{K}\) be the \(\mathbb Z_p^2\)-extension of \(K\), which is a \(\mathbb Z_p[[S, T]]\)-module in the usual way. If \(L_K\cap \widetilde{K}=K\), then \(\dim_{\mathbb F_p}(X_{ \widetilde{K}}/(p, S. T)) = \dim_{\mathbb F_p}(A_K/p)\). If \(L_K\cap \widetilde{K}\ne K\) and \(\dim_{\mathbb F_p}(A_K/p)=1\), there are two cases. If \(\lambda = 1\), then \(\dim_{\mathbb F_p}(X_{ \widetilde{K}}/(p, S. T))=1\). If \(\lambda \ge 2\), then \(\dim_{\mathbb F_p}(X_{ \widetilde{K}}/(p, S. T))=1\) if \(L_K\subset \widetilde{K}\) and \(=2\) otherwise. Under these assumptions, these yield necessary and sufficient conditions for \(X_{\widetilde{K}}\) to be a cyclic \(\mathbb Z_p[[S, T]]\)-module.
Reviewer: Lawrence C. Washington (College Park)On the generalized Brauer-Siegel theorem for asymptotically exact families of number fields with solvable Galois closurehttps://zbmath.org/1491.111032022-09-13T20:28:31.338867Z"Dixit, Anup B."https://zbmath.org/authors/?q=ai:dixit.anup-bAuthor's abstract: In 2002, \textit{M. A. Tsfasman} and \textit{S. G. Vlăduţ} [Mosc. Math. J. 2, No. 2, 329--402 (2002; Zbl 1004.11037)] formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable Galois closure.
Reviewer's remarks: To be explicit, the authors show the following result (combined as Theorem 3.1 and Theorem 3.2).
Theorem: Let \(T=\{K_i\}\) be either an asymptotic good tower of number fields, or an asymptotic bad family of number fields. Then the generalized Brauer-Siegel conjecture holds for \(T\) in case that each field \(K_i\) has solvable Galois closure over \(\mathbb{Q}\).
The notions in the theorem do require definitions, as follows. The set \(\{K_i\}\) is supposed to be sequence of number fields. Call it \(T\). \(T\) is said to be a family if \(K_i\ne K_j\) for \(i\ne j\). In addition, if \(K_i\subseteq K_{i+1}\) for all \(i\), then \(T\) is called a tower. A family \(T\) is said to be asymptotically exact if the limits \(\lim_{i\to\infty}\frac{r_j(K_i)}{g_{K_i}}\) with \(j=1\) and \(j=2\), and \(\lim_{\ell\to\infty} \frac{N_q(K_i)}{g_{K_i}}\) do exist for all prime powers \(q\); here \(g_{k_i}\) stands for \(\log\sqrt{\text{|discriminant}(K_i/\mathbb{Q})|}\) and \(r_1(K_i)\) and \(r_2(K_i)\) are the number of real and complex embeddings of \(K_i\), respectively, and \(N_q(K_i)\) denotes the number of non-Archimedean places \(\mu\) of \(K_i\) such that \(\operatorname{Norm }(\mu)= q\). An asymptotic exact family \(T=\{K_i\}\) is called asymptotically bad if all the above limits are equal to zero. And if an asymptotically exact family is not asymptotically bad, then it is called asymptotically good.
In the terminology by Tsfasman and Vlǎdut the generalized Brauer-Siegel conjecture is the following: \(\lim_{i\to\infty} \frac{\log h_{K_i}R_{K_i}}{g_{K_i}}\) exists, and it is equal to \[1+ \sum_q\left(\lim_{i\to\infty} \frac{N_q(K_i)}{g_{K_i}}\right)\left(\log\frac{q}{q-1}\right)- \left(\lim_{i\to\infty} \frac{r_1(K_i)}{g_{K_i}}\right)\cdot(\log 2)-\left(\lim_{i\to\infty} \frac{r_2(K_i)}{g_{k_i}}\right)\cdot(\log 2\pi);\] here \(R_{K_i}\) is the regulator of \(K_i\) and \(h_{K_i}\) is the class number of \(K_i\).
There are more properties involved here, and worked-out in the paper.
Reviewer: Robert W. van der Waall (Huizen)On the class numbers in the cyclotomic \(\mathbb Z_{29}\)- and \(\mathbb Z_{31}\)-extensions of the field of rationalshttps://zbmath.org/1491.111042022-09-13T20:28:31.338867Z"Kogoshi, Yuta"https://zbmath.org/authors/?q=ai:kogoshi.yuta"Morisawa, Takayuki"https://zbmath.org/authors/?q=ai:morisawa.takayukiFor a prime \(p\) and a non-negative integer \(n\), let \(\mathbb{B}_{p,n}\) be the unique real subfield of \(\mathbb{Q}(\zeta_{2p^{n+1}} )\) such that the Galois group of \(\mathbb{B}_{p,n}/\mathbb{Q}\) is isomorphic to the cyclic group \(\mathbb{Z}/{p^n\mathbb{Z}}\) of order \(p^n\). Further, \(h_{p, n}\) denotes the class number of \(\mathbb{B}_{p,n}\). In the paper under review, the authors consider the problem: Is the class number \(h_{p,n}\) indivisible by a given prime number \(\ell\)? Precisely, the authors prove that for \(p=29, 31\), \(h_{p,n}\) is indivisible by \(\ell\) provided \(\ell\) is a primitive root modulo \(p^2\).
Reviewer: Azizul Hoque (Allahabad)On Steinitz classes, realizable Galois module classes and embedding problems for non-abelian extensions of degree a power of 2https://zbmath.org/1491.111052022-09-13T20:28:31.338867Z"Mazhouda, Kamel"https://zbmath.org/authors/?q=ai:mazhouda.kamel"Sodaïgui, Bouchaïb"https://zbmath.org/authors/?q=ai:sodaigui.bouchaib"Taous, Mohammed"https://zbmath.org/authors/?q=ai:taous.mohammedLet \(k\) be a number field. For any finitely generated, torsion free \(O_k\)-module \(M\) of rank \(n\), there exists an ideal \(I\) of \(O_k\) such that \(M\simeq O^{n-1}_k \oplus I\), and the class of \(I\) in \(Cl(k)\) is called the Steinitz class of \(M\).
If \(\Gamma\) is a finite group, we can then consider the problem of realizable classes, which is the characterization of the set of ideal classes \(c\in Cl(k)\) such that there exists a tamely ramified extension \(N/k\), with Galois group isomorphic to \(\Gamma\) and where the Steinitz class of \(O_N\) is \(c\). The problem can be generalized to realizable Galois module classes (such as realization of classes in \(Cl(O_k[\Gamma])\)), and is well known when \(\Gamma\) is abelian (cf [\textit{L. R. McCulloh}, J. Reine Angew. Math. 375/376, 259--306 (1987; Zbl 0619.12008)]).
In this paper, following and completing results obtained in [\textit{F. Sbeity} and \textit{B. Sodaïgui}, Int. J. Number Theory 6, No. 8, 1769--1783 (2010; Zbl 1215.11110)], the authors consider cases where \(\Gamma\) is a non abelian group of order 16, or an extra-special group of order 32. They prove that given a 4-uple (or 6-uple) of elements in \(Cl(k)\), there exists 4 (or 6) quadratic tame extensions of \(k\), for which each Steinitz class realizes each of the given classes, and such that the compositum of those fields can be embedded in a tame \(\Gamma\)-extension of \(k\).
This result, and some consequences given, are a step in studying the more general problem of determining the structure of the Galois module classes realizable by the rings of integers of \(\Gamma\)-extensions.
Reviewer: Isabelle Dubois (Metz)A quaternionic construction of \(p\)-adic singular modulihttps://zbmath.org/1491.111062022-09-13T20:28:31.338867Z"Guitart, Xavier"https://zbmath.org/authors/?q=ai:guitart.xavier"Masdeu, Marc"https://zbmath.org/authors/?q=ai:masdeu.marc"Xarles, Xavier"https://zbmath.org/authors/?q=ai:xarles.xavier\textit{H. Darmon} and \textit{J. Vonk} [Duke Math. J. 170, No. 1, 23--93 (2021; Zbl 1486.11137)] had introduced rigid meromorphic cocycles as a conjectural \(p\)-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of \(\mathrm{SL}_2(\mathbb Z[1/p])\) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article, the authors present a construction of cohomology classes inspired by that of Darmon-Vonk, in which \(\mathrm{SL}_2(\mathbb Z[1/p])\) is replaced by an order in an indefinite quaternion algebra over a totally real number field \(F\). These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions \(K\) of \(F\), and they conjecture that the corresponding values lie in algebraic extensions of \(K\). They also report on extensive numerical evidence for this algebraicity conjecture.
Reviewer: Abdelmalek Azizi (Oujda)A modular relation involving non-trivial zeros of the Dedekind zeta function, and the generalized Riemann hypothesishttps://zbmath.org/1491.111072022-09-13T20:28:31.338867Z"Dixit, Atul"https://zbmath.org/authors/?q=ai:dixit.atul"Gupta, Shivajee"https://zbmath.org/authors/?q=ai:gupta.shivajee"Vatwani, Akshaa"https://zbmath.org/authors/?q=ai:vatwani.akshaaSummary: We give a number field analogue of a result of Ramanujan, Hardy and Littlewood, thereby obtaining a modular relation involving the non-trivial zeros of the Dedekind zeta function. We also provide a Riesz-type criterion for the Generalized Riemann Hypothesis for \(\zeta_{\mathbb{K}}(s)\). New elegant transformations are obtained when \(\mathbb{K}\) is a quadratic extension, one of which involves the modified Bessel function of the second kind.Moduli of local shtukas and Harris's conjecturehttps://zbmath.org/1491.111082022-09-13T20:28:31.338867Z"Hansen, David"https://zbmath.org/authors/?q=ai:hansen.davidThe paper under review deals with the cohomology of moduli space of mixed-characteristic local shtukas for \(\mathrm{GL}_n\). \textit{M. Rapoport} and \textit{Th. Zink} [Period spaces for \(p\)-divisible groups. Princeton, NJ: Princeton Univ. Press (1996; Zbl 0873.14039)] defined and studied the Rapopport-Zink spaces. These are local analogues of the Shimura varieties, and exist as rigid analytic varieties. As in the case of Shimura varieties, these spaces are expected to realize the local Langlands correspondance for reductive groups over local field. In his 2014 course at Berkeley, Scholze vastly generalized these spaces by constructing moduli spaces of mixed-characteristic local shtukas.
Let us fix a \(p\)-adic local field \(E\) with \([E: \mathbb{Q}_p]<\infty\). Let \(G\) be a split reductive group over \(E\) with fixed Borel \(\mathbf{B}\) over \(E\). Let \(\breve{E}\) be the completion of the maximal unramified extension of \(E\) and \(\mathbf{T}\subseteq \mathbf{B}\) be the maximal torus. We further fix a lift \(\sigma\in \text{Aut}(\breve{E}/E)\) of the \(q\)-Frobenius. The moduli space \(\text{Sht}_{G, \mu, b}\) depends on the group theoretic datum \((G, \mu, b)\), where \(\mu\in X_{\star}(\mathbf{T})_{\text{dom}}\) and \(b\in G(\breve{E})\) such that the \(\sigma\)-conjugacy class \([b]\in B(G)\) lies in the Kottwitz set \(B(G, \mu^{-1})\). This is a moduli space of local shtukas with one leg and infinite level structure. In general, the space \(\text{Sht}_{G, \mu, b}\) is not a rigid analytic variety and only a locally spatial diamond as defined in the Berkeley lectures on p-adic geometry. Note that a mixed-characteristic local shtukas can be interpreted as a \(G\)-bundle on the Fontaine-Fargues curve, which mimics the notion of a shtukas over function field by Drinfeld. The basic properties of \(\text{Sht}_{G, \mu, b}\) are summarized in Theorem 1.2 and proved in the case of \(G=\text{GL}_n\) in Section 2C.
Let \(C=\widehat{\bar{E}}\) and \(\text{ind}_{P}^G\) be the unnormalized parabolic induction. Following \textit{E. Mantovan} [Ann. Sci. Éc. Norm. Supér. (4) 41, No. 5, 671--716 (2008; Zbl 1236.11101)], the author considers the so called Hodge-Newton parabolic \(\mathbf{P}=\mathbf{M}\mathbf{U}\) associated to the datum \((G, \mu, b)\) such that \(\mathbf{M}\subsetneq G\). In this case, the datum \((G, \mu, b)\) is called Hodge-Newton reducible (Definition 1.3). The main result (Theorem 1.8) of the paper states that we have canonical \(G\)-equivariant isomorphisms
\[
H_c^i(\text{Sht}_{G, \mu, b}\times_{\text{Spd} \breve{E}}\text{Spd} C, \mathbb{Z}/\ell^n)\simeq \text{ind}_{\mathbf{P}}^G(H_c^{i-2d}(\text{Sht}_{\mathbf{M}, \mu, b}\times_{\text{Spd} \breve{E}}\text{Spd} C, \mathbb{Z}/\ell^n)(-d))
\]
for all \(i\geq 0\) compatible with all additional structures, where
\[
d=\text{dim} \text{Sht}_{G, \mu, b}-\text{dim} \text{Sht}_{\mathbf{M}, \mu, b}.
\]
In particular, these isomorphisms are compatible with the natural \(W_E\)-actions on both sides. This settles a conjecture of Harris (in the Hodge-Newton reducible case) saying that when \(b\) is not basic, no supercuspidal representation of \(G\) contributes to the Euler characteristic of \(H_c^i(\text{Sht}_{G, \mu, b}, \overline{\mathbb{\mathbb{Q}}}_\ell)\).
Finally, the case of general group \(G\) is also settled in a preprint by \textit{I. Gaisin} and \textit{N. Imai} [``Non-semi-stable loci in Hecke stacks and Fargues' conjecture'', Preprint, \url{arXiv:1608.07446}].
Reviewer: Taiwang Deng (Bonn)On the fixed points of an elliptic-curve version of self-power maphttps://zbmath.org/1491.111092022-09-13T20:28:31.338867Z"Shizuya, Hiroki"https://zbmath.org/authors/?q=ai:shizuya.hirokiSummary: Fixed points of the self-power map over a finite field have been studied in cryptology as a special case of modular exponentiation. In this note, we define an elliptic-curve version of the self-power map, enumerate the number of curves that contain at least one fixed point, and give its upper and lower bounds. Our result is a partial solution to the open question raised by \textit{L. Glebsky} and \textit{I. E. Shparlinski} in [Des. Codes Cryptography 56, No. 1, 35--42 (2010; Zbl 1213.11198)].A \(\log\)-\(\log\) speedup for exponent one-fifth deterministic integer factorisationhttps://zbmath.org/1491.111102022-09-13T20:28:31.338867Z"Harvey, David"https://zbmath.org/authors/?q=ai:harvey.david-i"Hittmeir, Markus"https://zbmath.org/authors/?q=ai:hittmeir.markusIn this paper, an algorithm for integer factorisation is presented. It factors rigorously and deterministically a positive integer \( N\) into primes in at most \[O\left(\frac{n^{1/5}(\log n)^{16/5}}{(\log\log n)^{3/5}} \right)\] bit operations. This result is an improvement on the previous best known result by a factor of \((\log \log N)^{3/5}\) [\textit{D. Harvey}, Math. Comput. 90, No. 332, 2937--2950 (2021; Zbl 1472.11311)]. The proof is reduced to the case that \(N\) is either prime or a product of two distinct primes, and then follows the same basic plan of the above paper with some modifications that lead to the \((\log \log N)^{3/5}\) speed up.
Reviewer: Dimitros Poulakis (Thessaloniki)Calculating ``small'' solutions of inhomogeneous relative Thue inequalitieshttps://zbmath.org/1491.111112022-09-13T20:28:31.338867Z"Gaál, István"https://zbmath.org/authors/?q=ai:gaal.istvanA relative Thue equation is one of the form \(N_{K/M}(X-\alpha Y)=m\), where: \(K/M\) is a (finite) extension of number-fields, \(N_{K/M}(\cdot)\) is the norm relative to this extension, \(X,Y\in\mathbb{Z}_M\) (= ring of integers of \(M\)), \(K=M(\alpha)\) with \(\alpha\) is an algebraic number of degree \(\geq 3\) over \(M\) and \(m\in\mathbb{Z}_M\). Such equations appeared in Siegel's work as early as 1921. An inhomogeneous relative Thue equation has the form \(N_{K/M}(X-\alpha Y+\lambda)=m\), where \(M,K,\alpha, m\) are as above and \(\lambda\in\mathbb{Z}_K\) may be either a variable or (as in this paper) fixed.
In the present paper the author is interested in effectively solving inhomogeneous relative Thue inequalities (obviously this study includes that of inhomogeneous relative Thue equations) which have the form
\[
|N_{K/M}(X-\alpha Y+\lambda)| \leq c_0Z^k, \quad Z=\max(\operatorname{house}{X},\operatorname{house}{Y})\leq Z_0
\]
where, for an algebraic number \(\gamma\), the symbol ``\(\operatorname{house}{\gamma}\)'' denotes the maximum of the absolute values of the conjugates of \(\gamma\), \(c_0\) is a given positive constant, \(k\geq 0\) is a fixed positive integer and \(Z_0\) is a given huge number, say of the size of \(10^{100}\). Actually, the author considers an even more general problem, special case of which is the above mentioned problem, namely, the following: Let \(\alpha_1,\ldots,\alpha_n\) be given non-zero distinct complex numbers and \(\lambda_1,\ldots,\lambda_n\) any given complex numbers with \(\max_j |\lambda_j| \leq c_{\lambda}\) (a given positive constant). Let also \(c_0>\) be a given constant and \(k\geq 0\) a given integer. Then, the author develops a method for the explicit solution of the inequality
\[
\left|\prod_{j=1}^n(X-\alpha_jY+\lambda_j)\right| \leq c_0Z^k, \quad X,Y\in\mathbb{Z}_M,\; Z=\max(\operatorname{house}{X}\,,\operatorname{house}{Y})\leq Z_0.
\]
His method is based on theorems proved in this paper (too technical to be reproduced in this review) and techniques developed in earlier works of him. Two explicit examples are discussed.
First example. Let \(\alpha_i\), \(i=1,\ldots,9\) the roots of \(x^9-9x^7+24x^5-2x^4-20x^3+3x^2+5x-1\) which is viewed as a polynomial over \(M=\mathbb{Q}(\sqrt{2})\), and \(\lambda_i=\alpha_i^2+2\alpha_i\) (\(1\leq i\leq 9\)). All solutions of
\[
\left|\prod_{j=1}^9(X-\alpha_jY+\lambda_j)\right| \leq10 \quad X,Y\in\mathbb{Z}_M,\; \max(\operatorname{house}{X}\,,\operatorname{house}{Y})\leq 10^{100}.
\]
are computed; it turns out that they are totally 138.
Second example. Let \(Μ=\mathbb{Q}(\sqrt{-2})\), \(f(t)=t^5-t^4-4t^3+3t^2+3t-1\in M[t]\). The problem is to compute all polynomials \(g(t)=t^2-Yt+X\in\mathbb{Z}_M[t]\) with \(\max(\operatorname{house}{X}\,,\operatorname{house}{Y})\leq 10^{100}\) and \(|\mathrm{Res}_M(f,g)|\leq 25\). According to this paper's arguments, this problem is reduced to solving \(|N_{K/M}(X-\alpha Y+\alpha^2)|\leq 25\) in \(X,Y\) as above, where \(K=\mathbb{Q}(\alpha,\sqrt{-2})\) and \(f(\alpha)=0\). According to the author's computations, there are exactly 39 solutions of the last inequality.
Remark. A misprint in the title ``How to apply Lemma 3?'' of Subsection 3.4: ``Lemma'' should be replaced by ``Theorem''.
Reviewer: Nikos Tzanakis (Iraklion)Adequate predimension inequalities in differential fieldshttps://zbmath.org/1491.120032022-09-13T20:28:31.338867Z"Aslanyan, Vahagn"https://zbmath.org/authors/?q=ai:aslanyan.vahagn-aThe notion of a ``predimension inequality'' (in the context of so-called \textit{Fraïssé construction}) has been introduced by Hrushovski in 1990s. In short, if there is a good predimension notion on a certain category of finite structures, then one can construct its Fraïssé limit (a special kind of a direct limit), which has good model-theoretical properties.
In the paper under review, the author considers predimension inequalities in differential fields and formalizes Zilber's notion of \textit{adequacy} of such an inequality. The main examples of predimension inequalities in this context are the Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the \(j\)-function (established by Pila and Tsimerman). The author shows (Theorem 1.3) that the Ax-Schanuel inequality for the \(j\)-function is adequate. Using this result, the author performs Hrushovski's construction and obtains a natural candidate for the first-order theory of the differential equation of the \(j\)-function.
Reviewer: Piotr Kowalski (Wroclaw)On length densitieshttps://zbmath.org/1491.130272022-09-13T20:28:31.338867Z"Chapman, Scott T."https://zbmath.org/authors/?q=ai:chapman.scott-thomas"O'Neill, Christopher"https://zbmath.org/authors/?q=ai:oneill.christopher"Ponomarenko, Vadim"https://zbmath.org/authors/?q=ai:ponomarenko.vadimThe set of lengths of an element \(x\) in a commutative monoid \(M\) and the elasticity for elements of monoid \(M\) along with the elasticity of monoid \(M\) itself have been well studied in the literature, particularly for Krull monoid, numerical monoid, Puiseux monoids and arithmetic congruence monoids. This paper discusses various new notions such as a length density \(\operatorname{LD}(x)\), asymptotic length density \(\overline{\operatorname{LD}}(x)\) of an element \(x\) and length density \(\operatorname{LD}(M)\) of a monoid \(M\). Analogous to the acceptance of elasticity of \(M\), the authors define the acceptance of length density. The length density of \(M\) is accepted if there exists \(x\in M\) such that \(\operatorname{LD}(x) = \operatorname{LD}(M)\).
The paper is organised as follows: Introduction includes a brief discussion of all the crucial definitions and notions which are required for the rest of the article. Section 2 discusses the basic properties of length density. In particularly, bounds are calculated for \(\operatorname{LD}(x)\) and \(\operatorname{LD}(M)\)s, and several examples of monoids have been given for which these bounds are met. These results further give examples of monoids for which length density is (is not) accepted. Section 3 answers the problem of the existence of a monoid having any irrational number in \((0, 1)\) as a length density, which is a similar result to the existence of a monoid having irrational elasticity in \((0, 1)\). However, this paper presents a new construction to prove the result for the length density case. Section 3 further computes the length density of block monoids and discusses several other examples. The last section of the article deals with the sufficient condition which guarantees the existence of asymptotic length density, this condition is particularly satisfied by a finitely generated monoid, \(C\)-monoid and Krull monoids with finite divisor class group. Authors also construct an example of a monoid which lacks asymptotic length density.
The article is well readable and has many good results with much information about the literature.
Reviewer: Nitin Bisht (Indore)On the algebraicity about the Hodge numbers of the Hilbert schemes of algebraic surfaceshttps://zbmath.org/1491.140072022-09-13T20:28:31.338867Z"Jin, Seokho"https://zbmath.org/authors/?q=ai:jin.seokho"Jo, Sihun"https://zbmath.org/authors/?q=ai:jo.sihunSummary: Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of \(n\) points of an algebraic surface is algebraic at a CM point \(\tau\) and rational numbers \(z_1\) and \(z_2\). Our result gives a refinement of the algebraicity on Betti numbers.Complete families of indecomposable non-simple abelian varietieshttps://zbmath.org/1491.140172022-09-13T20:28:31.338867Z"Flapan, Laure"https://zbmath.org/authors/?q=ai:flapan.laureThe main result of the paper shows how to construct complete families of indecomposable abelian varieties whose very general fiber is isogenous to a prescribed one and whose monodromy group is either a product of symplectic groups or a unitary group. The result applies to the study of the monodromy group of families of abelian varieties: it shows how to realize any product of symplectic groups of total rank g as the connected monodromy group of a complete family of $g'$-dimensional abelian varieties for any $g'\geq g$. This has a direct relation with the study of variations of Hodge structures on families of non-simple Jacobians, for instance Jacobians of coverings of curves. As a further application, it allows to recover certain Kodaira fibrations precedently found by the author with different methods and to construct a new one with fibers of genus 4.
Reviewer: Sara Torelli (Pavia)The wild McKay correspondence for cyclic groups of prime power orderhttps://zbmath.org/1491.140252022-09-13T20:28:31.338867Z"Tanno, Mahito"https://zbmath.org/authors/?q=ai:tanno.mahito"Yasuda, Takehiko"https://zbmath.org/authors/?q=ai:yasuda.takehikoSummary: The \(v\)-function is a key ingredient in the wild McKay correspondence. In this paper, we give a formula to compute it in terms of valuations of Witt vectors, when the given group is a cyclic group of prime power order. We apply it to study singularities of a quotient variety by a cyclic group of prime square order. We give a criterion whether the stringy motive of the quotient variety converges or not. Furthermore, if the given representation is indecomposable, then we also give a simple criterion for the quotient variety being terminal, canonical, log canonical, and not log canonical. With this criterion, we obtain more examples of quotient varieties which are Kawamata log terminal (klt) but not Cohen-Macaulay.Density of rational points on a family of del Pezzo surfaces of degree one (with an appendix by Jean-Louis Colliot-Thélène)https://zbmath.org/1491.140352022-09-13T20:28:31.338867Z"Desjardins, Julie"https://zbmath.org/authors/?q=ai:desjardins.julie"Winter, Rosa"https://zbmath.org/authors/?q=ai:winter.rosaThe paper under review concerns the density of rational points on certain del Pezzo surfaces of degree 1 over fields of characteristic zero, but really the motivation comes from the problem of unirationality over non-closed fields, say \(k\). Here, it is natural to restrict to \(k\)-minimal del Pezzo surfaces, and then unirationality is essentially open only for degree 1 surfaces. More precisely, the Picard number (over \(k\)) can only be 1 or 2 for these surfaces, and in the latter case unirationality follows from [\textit{J. Kollár} and \textit{M. Mella}, Am. J. Math. 139, No. 4, 915--936 (2017; Zbl 1388.14096)], but the Picard number 1 case seems completely open. It thus comes naturally to consider the problem of density of rational points of del Pezzo surfaces of degree 1 (and Picard number 1) as a test case, since this is clearly implied by unirationality. Previous (unconditional) results in this direction include [\textit{A. Várilly-Alvarado}, Algebra Number Theory 5, No. 5, 659--690 (2011; Zbl 1276.11114)] and [\textit{C. Salgado} and \textit{R. van Luijk}, Adv. Math. 261, 154--199 (2014; Zbl 1296.14018)], but there are also conditional results using root numbers by the first author in [J. Lond. Math. Soc., II. Ser. 99, No. 2, 295--331 (2019; Zbl 1459.11141)].
In detail, the authors consider the degree 6 surfaces \(S\) in weighted projective space \(\mathbb P[1,1,2,3]\) given by \[y^2 = x^3 + az^6+bz^3w^3+cw^3 \tag{1}\] where \(a,b,c\in k\) with \(ab\neq 0\) and \(4ac\neq b^2\) (the last condition is not stated explicitly in the paper, but together they ensure that the equations indeed define a del Pezzo surface of degree 1).
The main result states the sufficient condition that \(S(k)\) is Zariski dense if there are \(z,w\neq 0\) such that the elliptic curve resulting upon substituting into (1) has positive rank over \(k\). Moreover, this condition is also necessary if \(k\) is of finite type over \(\mathbb Q\).
To prove the density, the authors argue with the (isotrivial) elliptic surface defined by (1) (which has torsion-free Mordell-Weil group as a consequence of the above conditions). Formally, it is obtained from \(S\) by blowing up the base point \((0:0:1:1)\) of the linear system \(|-K_S|\). Starting from a non-trivial point \(R\) in a fibre of (1), the authors exhibit an auxilliary (possibly reducible) curve \(C_R\) which serves as trisection of the elliptic fibration. The curve is chosen as a special member of the linear system \(|-3K_S|\) which is singular at \(R\) and at two other points. It was pointed out by Kollàr that \(C_R\) can also be derived from a cubic surface which is \(3:1\) dominated by \(S\) in a natural way.
If the fibre \(F\) in question is smooth, then the authors prove that \(R\) can be chosen in such a way that
\begin{itemize}
\item \(C_R\) contains an irreducible component over \(k\) which gives a section of the fibration, or
\item \(C_R\) is geometrically integral of genus 0, or
\item \(C_R\) is geometrically integral of genus 1 for \(R\) varying over an open subset of \(F\).
\end{itemize}
The proof of the potential density in the main theorem then proceeds by a careful case-by-case analysis. The necessity of the given condition, over fields of finite type over \(\mathbb Q\), builds on a theorem Colliot-Thélène which improves upon Merel's theorem for bounding the torsion in families of elliptic curves.
The paper concludes with three interesting examples illustrating the techniques developed.
Reviewer: Matthias Schütt (Hannover)Diophantine stabilityhttps://zbmath.org/1491.140362022-09-13T20:28:31.338867Z"Mazur, Barry"https://zbmath.org/authors/?q=ai:mazur.barry"Rubin, Karl"https://zbmath.org/authors/?q=ai:rubin.karlLet \(K\) be a number field, \(\overline{ K}\) a separable closure of \(K\), \(V\) an irreducible variety over \(K\) and \(L\) a field containing \(K\). The authors say that \(V\) is {\em diophantine-stable for \(L/K\) } if \(V(L)=V(K)\). If \(\ell\) is a rational prime, they say that \(V\) is {\em \(\ell\)-diophantine stable} if for every positive integer \(n\), and every finite set \(\Sigma\) of places of \(K\), there are infinitely many cyclic extensions \(L/K\) of degree \(\ell^n\), completely split at all places \(v\in\Sigma\), such that \(V\) is diophantine-stable for \(L/K\). \par The first main result deals with a simple abelian variety \(A\) over \(K\). Assume that all \({\overline K}\)-endomorphisms of \(A\) are defined over \(K\). Then there is a set \(S\) of rational primes with positive density such that \(A\) is \(\ell\)-diophantine-stable over \(K\) for every \(\ell\in S\).
The second main result, which is a consequence of the first one, deals with an irreducible curve \(X\) over \(K\). Assume that the normalisation and completion \(\tilde X\) of \(X\) has genus \(\ge 1\) and that all \(\overline{K}\)-endomorphisms of the Jacobian of \(\tilde X\) are defined over \(K\); then there is a set \(S\) of rational primes with positive density such that \(X\) is \(\ell\)-diophantine-stable over \(K\) for every \(\ell\in S\).
From the second result the authors deduce the following two statements, the first one by applying repeatedly their result to the modular curve \(X_0(p)\), the second one to an elliptic curve over \({\mathbb Q}\) of positive rank. Let \(p\ge 23\) with \(p\not\in\{37,43,67,163\}\); then there are uncountably many pairwise non-isomorphic subfields \(L\) of \(\overline{\mathbb Q}\) such that no elliptic curve defined over \(L\) possesses an \(L\)-rational subgroup of ordre \(p\). And finally the authors prove that for every prime \(p\), there are uncountably many pairwise non-isomorphic totally real fields \(L\) of algebraic numbers in \({\mathbb Q}_p\) over which the following two statements both hold: (i) There is a diophantine definition of \(\mathbb Z\) in the ring of integers \({\mathcal O}_L\) of \(L\); in particular Hilbert's Tenth Problem has a negative answer for \({\mathcal O}_L\). (ii) There exists a first-order definition of the ring \(\mathbb Z\) in \(L\); the first-order theory for such fields \(L\) is undecidable.
The appendix by M. Larsen is devoted to the proof of the following result. Let \(A\) be a simple abelian variety defined over \(K\) such that \({\mathcal E}:={\mathrm{End}}_K(A)= {\mathrm{End}}_{\overline K}(A)\). Let \({\mathcal R}\) denote the center of \(\mathcal E\) and \({\mathcal M}={\mathcal R}\otimes{\mathbb Q}\). There is a positive density set \(S\) of rational primes such that for every prime \(\lambda\) of \({\mathcal M}\) lyning above \(S\) we have: (i) there is a \(\tau_0\in G_{K^{\mathrm ab}}\) such that \(A[\lambda]^{(\tau_0)}=0\), and (ii) there is a \(\tau_1\in G_{K^{\mathrm ab}}\) such that \(A[\lambda]/(\tau_1-1)A[\lambda]\) is a simple \({\mathcal E}/\lambda\)-module.
Reviewer: Michel Waldschmidt (Paris)Lifting low-dimensional local systemshttps://zbmath.org/1491.140402022-09-13T20:28:31.338867Z"De Clercq, Charles"https://zbmath.org/authors/?q=ai:de-clercq.charles"Florence, Mathieu"https://zbmath.org/authors/?q=ai:florence.mathieuSummary: Let \(k\) be a field of characteristic \(p>0\). Denote by \(\mathbf{W}_r(k)\) the ring of truntacted Witt vectors of length \(r \ge 2\), built out of \(k\). In this text, we consider the following question, depending on a given profinite group \(G\). \(\mathcal Q(G)\): Does every (continuous) representation \(G\longrightarrow \mathrm{GL}_d(k)\) lift to a representation \(G\longrightarrow \mathrm{GL}_d(\mathbf{W}_r(k))\)? We work in the class of cyclotomic pairs (Definition 4.3), first introduced in our paper [``Smooth profinite groups. I: geometrizing Kummer theory'', Preprint, \url{arXiv:2009.11130}] under the name ``smooth profinite groups''. Using Grothendieck-Hilbert' theorem 90, we show that the algebraic fundamental groups of the following schemes are cyclotomic: spectra of semilocal rings over \(\mathbb{Z}[\frac{1}{p}]\), smooth curves over algebraically closed fields, and affine schemes over \(\mathbb{F}_p\). In particular, absolute Galois groups of fields fit into this class. We then give a positive partial answer to \(\mathcal Q(G)\), for a cyclotomic profinite group \(G\): the answer is positive, when \(d=2\) and \(r=2\). When \(d=2\) and \(r=\infty \), we show that any 2-dimensional representation of \(G\) \text{stably} lifts to a representation over \(\mathbf{W}(k)\): see Theorem 6.1. When \(p=2\) and \(k=\mathbb{F}_2\), we prove the same results, up to dimension \(d=4\). We then give a concrete application to algebraic geometry: we prove that local systems of low dimension lift Zariski-locally (Corollary 6.3).Arithmetic intersections of modular geodesicshttps://zbmath.org/1491.140412022-09-13T20:28:31.338867Z"Darmon, Henri"https://zbmath.org/authors/?q=ai:darmon.henri"Vonk, Jan"https://zbmath.org/authors/?q=ai:vonk.janThe authors write: ``The goal of this note is to propose an arithmetic intersection theory for modular geodesics, attaching to a pair of such geodesics certain numerical invariants that are rich enough to (ostensibly) generate class fields of real quadratic fields. The predicted algebraicity of these quantities is a by-product of the approach of \textit{H. Darmon} and \textit{J. Vonk} [Duke Math. J. 170, No. 1, 23--93 (2021; Zbl 1486.11137)] to explicit class field theory based on the RM values of rigid meromorphic cocycles, but avoids the latter notion and offers a somewhat complementary perspective.''
Here ``RM'' stands for real multiplication and a ``modular geodesic'' is the image on \(\mathrm{SL}_{2}(\mathbb{Z})\setminus\mathfrak{H}\) of the oriented geodesic joining the two roots of an indefinite primitive integral binary quadratic form on the upper half-plane \(\mathfrak{H}\).
For any pair of distinct closed geodesics \((\gamma_{1}, \gamma_{2})\) on \(\Gamma\setminus\mathfrak{H}\), where \(\Gamma\) is an arithmetic subgroup of \(\mathrm{SL}_{2}(\mathbb{R})\) arising from the multiplicative group of an order in an indefinite quaternion algebra \(B\) over \(\mathbb{Q}\), the authors define an arithmetic intersection \(\gamma_{1} \star \gamma_{2}\). Given a rational prime \(p\) not dividing the discriminant of \(B\) and an order \(R\) of \(B\) of discriminant prime to \(p\), let \(\Gamma_{p}:=(R[1/p])_{1}^{\star}\) (here \(O_{1}^{\star}\) stands for the group of the elements of an order \(O\) of reduced norm one), then a \(p\)-arithmetic intersection \((\gamma_{1} \star \gamma_{2})_{\Gamma_{p}}\) is defined as a certain infinite \(p\)-adically absolutely convergent product.
At last conjecturally certain ``incoherent intersection numbers'' attached to a pair of RM divisors are defined and their conjectural relation to compatible systems of geodesics on an ``incoherent collection'' of Shimura curves is briefly discussed. The authors state seven conjectures and ask a few questions concerning the hypothetical algebraic properties of the defined intersections.
Reviewer: B. Z. Moroz (Bonn)Application of automorphic forms to lattice problemshttps://zbmath.org/1491.140422022-09-13T20:28:31.338867Z"Düzlü, Samed"https://zbmath.org/authors/?q=ai:duzlu.samed"Krämer, Juliane"https://zbmath.org/authors/?q=ai:kramer.julianeThe paper under review is motivated by the work of de Boer, Ducas, Pellet-Mary, and Wesolowski on self-reducibility of ideal-SVP via Arakelov random walks [\textit{D. Micciancio} (ed.) and \textit{T. Ristenpart} (ed.), Advances in cryptology -- CRYPTO 2020. 40th annual international cryptology conference, CRYPTO 2020. Proceedings. Part II. Cham: Springer. 243--273 (2020; Zbl 07332292)]. More preceisely, the authors show a worst-case to average-case reduction for ideal lattices and explain their approach how the steps are reproduced for module lattices of a fixed rank over some number field. Two major distinctions in their approach are that for higher rank module lattices, the notion of Arakelov divisors is replaced by adèles and Fourier analysis is substituted by the notion of automorphic forms.
Note that subject to the Riemann hypothesis, the worst-case to average-case convergence is analyzed in terms of the Fourier series. Thereafter, the worst-case shortest vector problem is as hard as the averagecase shortest vector problem.
Reviewer: Sami Omar (Sukhair)Elliptic curve involving subfamilies of rank at least 5 over \(\mathbb{Q}(t)\) or \(\mathbb{Q}(t,k)\)https://zbmath.org/1491.140522022-09-13T20:28:31.338867Z"Youmbai, Ahmed El Amine"https://zbmath.org/authors/?q=ai:youmbai.ahmed-el-amine"Uludağ, Muhammed"https://zbmath.org/authors/?q=ai:uludag.muhammed-a"Behloul, Djilali"https://zbmath.org/authors/?q=ai:behloul.djilaliThe authors provide one- and two-parameter families of elliptic curves with large Mordell-Weil rank. In fact, they exhibit elliptic curves over \(\mathbb Q(t)\) with rank at least 5 that are either induced by the edges of a rational cuboid or by Diophantine triples. Using careful specialization of \(t\), examples of elliptic curves over \(\mathbb Q\) with rank 8, 9, 10 and 11 are given.
Reviewer: Mohammad Sadek (New Cairo)Computing discrete invariants of varieties in positive characteristic. I: Ekedahl-Oort types of curveshttps://zbmath.org/1491.140652022-09-13T20:28:31.338867Z"Moonen, Ben"https://zbmath.org/authors/?q=ai:moonen.benSummary: We develop a method to compute the Ekedahl-Oort type of a curve \(C\) over a field \(k\) of characteristic \(p\) (which is the isomorphism type of the \(p\)-kernel group scheme \(J[p]\), where \(J\) is the Jacobian of \(C)\). Part of our method is general, in that we introduce the new notion of a Hasse-Witt triple, which re-encodes in a useful way the information contained in the Dieudonné module of \(J[p]\). For complete intersection curves we then give a simple method to compute this Hasse-Witt triple. An implementation of this method is available in Magma.Zero-sum triangles for involutory, idempotent, nilpotent and unipotent matriceshttps://zbmath.org/1491.150342022-09-13T20:28:31.338867Z"Hao, Pengwei"https://zbmath.org/authors/?q=ai:hao.pengwei"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.7"Hao, Huahan"https://zbmath.org/authors/?q=ai:hao.huahanThe authors present the zero-sum rule
\[
t(i,k-1)+t(i,k)+t(i+1,k)=0
\]
for recurrence relations to construct integer triangles as triangular matrices with involutory, idempotent, nilpotent, and unipotent properties.
Reviewer: Jorma K. Merikoski (Tampere)Max and min matrices with hyper-Fibonacci numbershttps://zbmath.org/1491.150352022-09-13T20:28:31.338867Z"Solmaz, Tuğçe"https://zbmath.org/authors/?q=ai:solmaz.tugce"Bahşi, Mustafa"https://zbmath.org/authors/?q=ai:bahsi.mustafaOn a conjecture about an analogue of Tokuyama's theorem for \(G_2\)https://zbmath.org/1491.170042022-09-13T20:28:31.338867Z"DeFranco, Mario"https://zbmath.org/authors/?q=ai:defranco.mario\textit{T. Tokuyama} [J. Math. Soc. Japan 40, No. 4, 671--685 (1988; Zbl 0639.20022)] expressed the product of the deformed Weyl denominator and a Schur polynomial as a sum over strict Gelfand-Tsetlin patterns. This may be recast using crystal graphs as follows (see [\textit{B. Brubaker} et al., Weyl group multiple Dirichlet series. Type A combinatorial theory. Princeton, NJ: Princeton University Press (2011; Zbl 1288.11052)], Chapter 5). Let \(\lambda\in \mathfrak{sl}_{r+1}(\mathbb{C})\) be a dominant weight, \(\chi_\lambda\) the character of the associated representation of highest weight \(\lambda\), and \(\rho\) be the Weyl vector. Then Tokuyama's formula is an expression for the product of \(\chi_\lambda\) with the deformed Weyl denominator as a sum over \(\mathcal{B}_{\lambda+\rho}\), the Kashiwara crystal with highest weight \(\lambda+\rho\). There is a substantial history of this formula for other Cartan types, explained in the paper under review. In particular, \textit{H. Friedlander} et al. [J. Algebr. Comb. 41, No. 4, 1089--1102 (2015; Zbl 1316.05128)] conjectured such a formula for the exceptional Lie algebra of type \(G_2\). The paper under review establishes this conjecture using combinatorial methods. The author expresses both sides of the conjectured identity as polynomials in four variables whose coefficients are rational functions, and then shows that these coefficients are equal.
A complement to this work is the paper of \textit{S. Leslie} [Sel. Math., New Ser. 25, No. 3, Paper No. 41, 50 p. (2019; Zbl 1491.17005)], which gives a different expression in type \(G_2\) for the character times the deformed Weyl denominator, namely a sum over \(\mathcal{B}_{\lambda+\rho}\) plus a geometric error term coming from finitely many crystals of resonant Mirković-Vilonen polytopes.
Reviewer: Solomon Friedberg (Chestnut Hill)Groups containing locally maximal product-free sets of size 4https://zbmath.org/1491.200582022-09-13T20:28:31.338867Z"Anabanti, C. S."https://zbmath.org/authors/?q=ai:anabanti.chimere-stanleySummary: Every locally maximal product-free set \(S\) in a finite group \(G\) satisfies \(G=S\cup SS \cup S^{-1}S \cup SS^{-1}\cup \sqrt{S}\), where \(SS=\{xy\mid x, y\in S\}\), \(S^{-1}S=\{x^{-1}y\mid x, y\in S\}\), \(SS^{-1}=\{xy^{-1}\mid x, y\in S\}\) and \(\sqrt{S}=\{x\in G\mid x^2\in S\}\). To better understand locally maximal product-free sets, \textit{E. A. Bertram} [Discrete Math. 44, 31--43 (1983; Zbl 0506.05060)] asked whether every locally maximal product-free set \(S\) in a finite abelian group satisfy \(|\sqrt{S}|\leqslant 2|S|\). This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size \(4\), continuing the work of \textit{A. P. Street} and \textit{E. G. Whitehead jun.} [J. Comb. Theory, Ser. A 17, 219--226 (1974; Zbl 0288.05020); Lect. Notes Math. 403, 109--124 (1974; Zbl 0318.05010)] and \textit{M. Giudici} and \textit{S. Hart} [Electron. J. Comb. 16, No. 1, Research Paper R59, 17 p. (2009; Zbl 1168.20009)] on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size \(4\), and conclude with a conjecture on the size \(4\) problem as well as an open problem on the general case.The Fibonacci-circulant sequences in the binary polyhedral groupshttps://zbmath.org/1491.200772022-09-13T20:28:31.338867Z"Karaduman, Erdal"https://zbmath.org/authors/?q=ai:karaduman.erdal"Deveci, Omur"https://zbmath.org/authors/?q=ai:deveci.omurSummary: In 2017 \textit{Ö. Deveci} et al. [Iran. J. Sci. Technol., Trans. A, Sci. 41, No. 4, 1033--1038 (2017; Zbl 1391.11026)] defined the Fibonacci-circulant sequences of the first and second kinds as shown, respectively:
\[
x_n^1 = -x_{n-1}^1+x_{n-2}^1-x_{n-3}^1\text{ for }n \geq 4,\text{ where } x_1^1=x_2^1=0\text{ and } x_3^1=1
\]
and
\[
x_n^2 = -x_{n-3}^2-x_{n-4}^2+x_{n-5}^2\text{ for }n \geq 6, \text{ where }x_1^2 = x_2^2 = x_3^2=x_4^2=0\text{ and }x_5^2=1
\]
Also, they extended the Fibonacci-circulant sequences of the first and second kinds to groups. In this paper, we obtain the periods of the Fibonacci-circulant sequences of the first and second kinds in the binary polyhedral groups.Frobenius R-variety of the numerical semigroups contained in a given onehttps://zbmath.org/1491.201212022-09-13T20:28:31.338867Z"Rosales, J. C."https://zbmath.org/authors/?q=ai:rosales.jose-carlos"Branco, M. B."https://zbmath.org/authors/?q=ai:branco.manuel-baptista|branco.manuel-batista"Traesel, M. A."https://zbmath.org/authors/?q=ai:traesel.marcio-andreSummary: Let \(\triangle\) be a numerical semigroup and \(\mathrm{R}(\triangle)=\{S \mid S \text{ is a numericalsemigroup and } S\subseteq \triangle\}\). We prove that \(\mathrm{R}(\triangle)\) is Frobenius R-variety that can be arranged in a tree rooted in \(\triangle\). We introduce the concepts of Frobenius and genus number of \(S\) restricted to \(\triangle\) (respectively \(\mathrm{F}_{\triangle}(S)\) and \(\mathrm{g}_{\triangle}(S)\)). We give formulas for \(\mathrm{F}_{\triangle}(S)\), \(\mathrm{g}_{\triangle}(S)\) and generalizations of the Amorós's and Wilf's conjecture. Moreover, we will show that most of the results of irreducibility can be generalized to \(\mathrm{R}(\triangle)\)-irreducibility.On certain \(\mathrm{Sp}\)-distinguished principal series representations of the quasi-split unitary groupshttps://zbmath.org/1491.220052022-09-13T20:28:31.338867Z"Mitra, Arnab"https://zbmath.org/authors/?q=ai:mitra.arnab|mitra.arnab.1This paper studies distinguished representations of a quasi-split unitary group \(U_{2n}\) over a non-archimedean local field of odd residue characteristic. Let \(F\) be such a local field and \(E\) a quadratic extension of \(F\). Then the symplectic group \(\mathrm{Sp}_{2n}(F)\) is a subgroup of \(U_{2n}(E)\) fixed under an involution on \(U_{2n}(E)\). Dijols-Prasad made conjecture that for \(L-\)packets associated to Arthur packets on \(U_{2n}(F)\), the distinction of representations in packet \(\{\pi\}\) with respect to \(\mathrm{Sp}_{2n}(F)\) is equivalent to the distinction of the base change \(BC(\pi)\) to \(\mathrm{GL}_{2n}(E)\) with respect to \(\mathrm{Sp}_{2n}(E)\). The paper verifies the conjecture in case \(n=2m\) and \(\pi\) is an induced representation from unitarizable Speh representation of \(\mathrm{GL}_{2m}(E)\) (considered as a Levi subgroup of \(U_{2n}(E))\).
The paper establishes a more general result, for induced representation of the form \(\pi\rtimes \sigma\) where \(\sigma\) is a cuspidal representation of \(U_{2k}(E)\) and \(\pi\) is a unitarizable Speh representation of \(\mathrm{GL}_{l}(E)\). Such representation is distinguished if and only if \(k=0\) and \(\pi\) is \(\mathrm{Sp}\)-distinguished (in particular \(l\) is even). The proof uses the geometric lemma by Bernstein-Zelevinsky.
Reviewer: Zhengyu Mao (Newark)On functional equations of degenerate dilogarithmhttps://zbmath.org/1491.390132022-09-13T20:28:31.338867Z"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san"Lee, Hyunseok"https://zbmath.org/authors/?q=ai:lee.hyunseok"Kwon, Jongkyum"https://zbmath.org/authors/?q=ai:kwon.jongkyum"Kim, Yunjae"https://zbmath.org/authors/?q=ai:kim.yunjaeSummary: Recently, the degenerate polylogarithm is introduced by \textit{T. Kim} and \textit{D. S. Kim} [Proc. Jangjeon Math. Soc. 25, No. 1, 1--11 (2022; Zbl 07539984)]
as a degenerate version of the polylogaritmn. In this note, we derive some interesting functional equations related to degenerate dilogarithm.\( \mu \)-statistical convergence and the space of functions \(\mu \)-stat continuous on the segmenthttps://zbmath.org/1491.400032022-09-13T20:28:31.338867Z"Sadigova, S. R."https://zbmath.org/authors/?q=ai:sadigova.sabina-rahibThe author introduces $\mu$-statistical density of a point and the concept of $\mu$-statistical fundamentality at a point and also states that this method is equivalent to the concept of $\mu$-stat convergence. On the other hand, the concept of $\mu$-stat continuity is defined. Some properties of the space of all $\mu$-stat continuous functions are examined.
Reviewer: Emre Taş (Kırşehir)Tracial moment problems on hypercubeshttps://zbmath.org/1491.440062022-09-13T20:28:31.338867Z"Le, Cong Trinh"https://zbmath.org/authors/?q=ai:le-cong-trinh.Summary: In this paper we introduce the \textit{tracial \(K\)-moment problem} and the \textit{sequential matrix-valued \(K\)-moment problem} and show the equivalence of the solvability of these problems. Using a Haviland's theorem for matrix polynomials, we solve these \(K\)-moment problems for the case where \(K\) is the hypercube \([-1,1]^n\).Quaternionic product of circles and cycles and octonionic product for pairs of circleshttps://zbmath.org/1491.510162022-09-13T20:28:31.338867Z"Crasmareanu, Mircea"https://zbmath.org/authors/?q=ai:crasmareanu.mirceaThis paper, computational in nature, introduces some products on the set of circles, cycles and spheres, and then suggests some applications, some with a hyperbolic flavor, and one to Euclidean geometry.
Reviewer: Victor V. Pambuccian (Glendale)Ruled surfaces in Minkowski 3-space and split quaternion operatorshttps://zbmath.org/1491.530062022-09-13T20:28:31.338867Z"Aslan, Selahattin"https://zbmath.org/authors/?q=ai:aslan.selahattin"Bekar, Murat"https://zbmath.org/authors/?q=ai:bekar.murat"Yaylı, Yusuf"https://zbmath.org/authors/?q=ai:yayli.yusufThis article is a direct extension of [\textit{S. Aslan} et al., J. Geom. Phys. 161, Article ID 104048, 10 p. (2021; Zbl 1460.53004)]. Similar to the techniques given in [loc. cit.], here the authors define an operator, called split quaternion operator \(Q(s, t) = s + u(t)\). Here \(s\) be a real variable and \(u(t)\) is a curve in the Lorentz Minkowski space \(\mathbb E_1^3\). The authors classify this operator into three types based on the norm of the quaternion.
Let \(w(t)\) be the curve on the Lorentzian unit sphere or on the hyperbolic unit sphere, then one of the results in the article says that the split quaternion product of the split quaternion operator \(Q(s, t)\) and the pure split quaternion \(w(t)\) yields a ruled surface in \(\mathbb E_1^3\) if the position vector of the spherical curve \(w(t)\) is perpendicular to the vector part of the operator \(Q(s, t)\).
Reviewer: Pradip Kumar (Lucknow)A note on discrete degenerate random variableshttps://zbmath.org/1491.600162022-09-13T20:28:31.338867Z"Kim, Taekyun"https://zbmath.org/authors/?q=ai:kim.taekyun"Kim, Dae San"https://zbmath.org/authors/?q=ai:kim.dae-san"Jang, Lee-Chae"https://zbmath.org/authors/?q=ai:jang.lee-chae|jang.leechae"Kim, H. Y."https://zbmath.org/authors/?q=ai:kim.hwan-y|kim.ho-yeun|kim.heon-young|kim.hae-young|kim.hyun-young|kim.ha-young|kim.hye-yong|kim.hyun-yul|kim.ho-youn|kim.hun-young|kim.ho-yon|kim.hae-yong|kim.hye-yeonSummary: In this paper, we introduce two discrete degenerate random variables, namely the degenerate binomial and degenerate Poisson random variables. We deduce the expectations of the degenerate binomial random variables. We compute the generating function of the moments of the degenerate Poisson random variables, which leads us to define the new type degenerate Bell polynomials, and hence obtain explicit expressions for the moments of those random variables in terms of such polynomials. We also get the variances of the degenerate Poisson random variables. Finally, we illustrate two examples of the degenerate Poisson random variables.Periods of iterations of functions with restricted preimage sizeshttps://zbmath.org/1491.600172022-09-13T20:28:31.338867Z"Martins, Rodrigo S. V."https://zbmath.org/authors/?q=ai:martins.rodrigo-s-v"Panario, Daniel"https://zbmath.org/authors/?q=ai:panario.daniel"Qureshi, Claudio"https://zbmath.org/authors/?q=ai:qureshi.claudio-m"Schmutz, Eric"https://zbmath.org/authors/?q=ai:schmutz.ericYet another way of calculating moments of the Kesten's distribution and its consequences for Catalan numbers and Catalan triangleshttps://zbmath.org/1491.600322022-09-13T20:28:31.338867Z"Szabłowski, Paweł J."https://zbmath.org/authors/?q=ai:szablowski.pawel-jerzySummary: We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, also concerning Lucas, Fibonacci and Fine numbers.More on change-making and related problemshttps://zbmath.org/1491.681262022-09-13T20:28:31.338867Z"Chan, Timothy M."https://zbmath.org/authors/?q=ai:chan.timothy-m-y"He, Qizheng"https://zbmath.org/authors/?q=ai:he.qizhengThis paper considers the change-making or coin-changing problem: given a set of \(n\) types of coins, each of integer value, the goal is to select the minimum number of coins whose total value is equal to a specified target value \(t\). This problem is known to be weakly NP-hard and the previously to this paper best known algorithm for it had time complexity \(O(t \log t \log \log t)\).
The paper also considers the all-targets version of the problem, where the goal is to solve the change-making problem for all target values \(j\) between 1 and a specified upper bound \(t\). The previously to this paper best algorithm for the problem has time complexity \(\tilde O(t^{3/2})\), where as usual the \(\tilde O\) notation hides polylogarithmic factors.
The paper gives a simple FFT-based algorithm for the all-targets version of the problem that has time complexity \(\tilde O(t^{4/3})\). The complexity of the problem is also studied with respect to the largest coin value \(u\), and an algorithm is given for the problem with time complexity \(O(u^2 \log u + t)\). This is a very simple 3-line algorithm that interestingly has a rather complex correctness proof using a theorem on the Frobenius problem. This algorithm can be modified to solve the all-capacities unbounded knapsack problem within the same time bound.
The final results presented in the paper are
\begin{itemize}
\item an \(\tilde O((t \sigma)^{2/3}+t)\) algorithm for the all-targets change-making problem, where \(\sigma\) is the sum of values of the \(n\) coin types
\item an \(\tilde O(u)\) algorithm for the change-making problem
\item an \(\tilde O(nu)\) algorithm for the unbounded knapsack problem, and
\item a proof that an algorithm of Bringmann et al. can solve the minimum word-break problem in time \(\tilde O(nm^{1/3}+m)\), where the minimum word-break problem consists in expressing a string \(s\) of length \(n\) as the concatenation of the smallest number of words from a set \(D\) of strings with total length \(m\).
\end{itemize}
Reviewer: Roberto Solis-Oba (London)Restriction for general linear groups: the local non-tempered Gan-Gross-Prasad conjecture (non-Archimedean case)https://zbmath.org/1491.810212022-09-13T20:28:31.338867Z"Chan, Kei Yuen"https://zbmath.org/authors/?q=ai:chan.kei-yuenSummary: We prove a local Gan-Gross-Prasad conjecture on predicting the branching law for the non-tempered representations of general linear groups in the case of non-Archimedean fields. We also generalize to Bessel and Fourier-Jacobi models and study a possible generalization to Ext-branching laws.Generalization of the 2-form interactionshttps://zbmath.org/1491.830382022-09-13T20:28:31.338867Z"Heisenberg, Lavinia"https://zbmath.org/authors/?q=ai:heisenberg.lavinia"Trenkler, Georg"https://zbmath.org/authors/?q=ai:trenkler.georg(no abstract)The EFT likelihood for large-scale structurehttps://zbmath.org/1491.850032022-09-13T20:28:31.338867Z"Cabass, Giovanni"https://zbmath.org/authors/?q=ai:cabass.giovanni"Schmidt, Fabian"https://zbmath.org/authors/?q=ai:schmidt.fabian(no abstract)Attacking the linear congruential generator on elliptic curves via lattice techniqueshttps://zbmath.org/1491.940532022-09-13T20:28:31.338867Z"Gutierrez, Jaime"https://zbmath.org/authors/?q=ai:gutierrez.jaimeSummary: In this paper we study the linear congruential generator on elliptic curves from the cryptographic point of view. We show that if sufficiently many of the most significant bits of the composer and of three consecutive values of the sequence are given, then one can recover the seed and the composer (even in the case where the elliptic curve is private). The results are based on lattice reduction techniques and improve some recent approaches to the same security problem. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for nonlinear congruential generators. Several examples are tested using implementations of our algorithms.On the boomerang uniformity of a class of permutation quadrinomials over finite fieldshttps://zbmath.org/1491.940722022-09-13T20:28:31.338867Z"Wu, Yanan"https://zbmath.org/authors/?q=ai:wu.yanan"Wang, Lisha"https://zbmath.org/authors/?q=ai:wang.lisha"Li, Nian"https://zbmath.org/authors/?q=ai:li.nian"Zeng, Xiangyong"https://zbmath.org/authors/?q=ai:zeng.xiangyong"Tang, Xiaohu"https://zbmath.org/authors/?q=ai:tang.xiaohuSummary: Let \(\mathbb{F}_{2^n}\) be a finite field with \(2^n\) elements and \(f_{c_{\_}}(x) = c_0 x^{2^m ( 2^k + 1 )} + c_1 x^{2^{m + k} + 1} + c_2 x^{2^m + 2^k} + c_3 x^{2^k + 1} \in \mathbb{F}_{2^n} [x]\), where \(n\), \(m\) and \(k\) are positive integers with \(n = 2 m\) and \(\gcd(m, k) = e\). In this paper, motivated by a recent work of , we further study the boomerang uniformity of \(f_{c_{\_}}(x)\) by using similar ideas and carrying out particular techniques in solving equations over finite fields. As a consequence, we generalize Li, Xiong and Zeng's result [loc. cit.] from the case of \(m\) being odd and \(e = 1\) to that of both \(m / e\) and \(k / e\) being odd.Additive and linear conjucyclic codes over \(\mathbb{F}_4\)https://zbmath.org/1491.940842022-09-13T20:28:31.338867Z"Abualrub, Taher"https://zbmath.org/authors/?q=ai:abualrub.taher-a"Dougherty, Steven T."https://zbmath.org/authors/?q=ai:dougherty.steven-tIn this paper, the authors are interested in linear and additive conjucyclic codes over the finite field \(\mathbb{F}_4\). There is produced a special kind of duality for which the orthogonal (with respect to that duality) of conjucyclic codes is conjucyclic. Moreover, there is shown that this is not the case for other standard dualities. The authors prove that additive conjucyclic codes are the only non-trivial conjucyclic codes over \(\mathbb{F}_4\) and they classify these codes by linear algebraic approach. Finally, there is shown that additive conjucyclic codes of length \(n\) over \(\mathbb{F}_4\) are isomorphic to binary cyclic codes of length \(2n\).
Reviewer: Zlatko Varbanov (Veliko Tarnovo)Codes from unit groups of division algebras over number fieldshttps://zbmath.org/1491.940852022-09-13T20:28:31.338867Z"Maire, Christian"https://zbmath.org/authors/?q=ai:maire.christian"Page, Aurel"https://zbmath.org/authors/?q=ai:page.aurelLenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, \textit{C. Maire} and \textit{F. Oggier} [J. Pure Appl. Algebra 227 (7), 1827--1858 (2018; Zbl 1430.94101)] generalized these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyze the resulting codes.
In the current paper, it is shown that for all \(d \ge 2\), there exists a family of asymptotically good number field codes for the sum-rank distance, each obtained from the group of units of reduced norm 1 in a maximal order in a division algebra of degree \(d\), over a fixed alphabet \(M_d(\mathbb{F}_p)\), where \(\log p = c \log d + O(\log\log d)\) and \(c > 0\) is a constant.
It is also shown that for all \(d \ge 2\), there exists a family of asymptotically good number field codes for the sum-rank distance, each obtained from the additive group of a maximal order in a division algebra of degree \(d\), over a fixed alphabet \(M_d(\mathbb{F}_p)\), where \(\log p = \frac{1}{2} \log d + O(\log\log d)\).
Reviewer: Kanat Abdukhalikov (Al Ain)``Denominate numbers'' in mathematics school textbooks by Stefan Banachhttps://zbmath.org/1491.970022022-09-13T20:28:31.338867Z"Karpińska, Karolina"https://zbmath.org/authors/?q=ai:karpinska.karolinaSummary: The paper is dedicated to analysing the role of ``denominate numbers'' in textbooks for Polish schools, whose author or co-author was Stefan Banach. Banach used the concepts of ``number'' and ``denominate number'' in his textbooks. The ways of introducing these concepts, as well as the associated manner of understanding numeration, are analysed in the paper. The methods of manipulating ``denominate numbers'' through various procedures are also discussed. The analysis is carried out in the context of the Polish ministerial ordinances related to teaching mathematics and the content of other textbooks, especially those used by Banach as a secondary school student.Using sona to calculate the greatest common divisor of two integershttps://zbmath.org/1491.970032022-09-13T20:28:31.338867Z"Barros, Pedro Henrique Alves"https://zbmath.org/authors/?q=ai:barros.pedro-henrique-alves"da Silva, Patrícia Nunes"https://zbmath.org/authors/?q=ai:da-silva.patricia-nunes(no abstract)Investigating Catalan numbers with Pascal's trianglehttps://zbmath.org/1491.970042022-09-13T20:28:31.338867Z"Hong, Dae S."https://zbmath.org/authors/?q=ai:hong.dae-s(no abstract)Remainder and quotient without polynomial long divisionhttps://zbmath.org/1491.970052022-09-13T20:28:31.338867Z"Laudano, Francesco"https://zbmath.org/authors/?q=ai:laudano.francescoSummary: We propose an algorithm that allows calculating the remainder and the quotient of division between polynomials over commutative coefficient rings, without polynomial long division. We use the previous results to determine the quadratic factors of polynomials over commutative coefficient rings and, in particular, to completely factorize in \(\mathbb Z[x]\) any integral polynomial with degree less than 6. The arguments are suitable for building classroom/homework activities in basic algebra courses.A forward approach for solving linear Diophantine equationhttps://zbmath.org/1491.970062022-09-13T20:28:31.338867Z"Man, Yiu-Kwong"https://zbmath.org/authors/?q=ai:man.yiu-kwongSummary: A simple forward approach for solving linear Diophantine equations is presented, which does not involve using backward substitutions. It is suitable for teaching undergraduate students as an alternative to the backward substitution method commonly described in mathematics textbooks.Sequences of ratios of a convex quadrilateralhttps://zbmath.org/1491.970072022-09-13T20:28:31.338867Z"Laudano, Francesco"https://zbmath.org/authors/?q=ai:laudano.francesco(no abstract)