Recent zbMATH articles in MSC 11N https://zbmath.org/atom/cc/11N 2022-06-24T15:10:38.853281Z Werkzeug On composite odd numbers $$k$$ for which $$2^n k$$ is a noncototient for all positive integers $$n$$ https://zbmath.org/1485.11014 2022-06-24T15:10:38.853281Z "González, Marcos J." https://zbmath.org/authors/?q=ai:gonzalez.marcos-jose "Mendoza, Alberto" https://zbmath.org/authors/?q=ai:mendoza.alberto "Luca, Florian" https://zbmath.org/authors/?q=ai:luca.florian "Mejía Huguet, V. Janitzio" https://zbmath.org/authors/?q=ai:mejia-huguet.v-janitzio An odd positive integer $$k$$ is a \textit{Riesel number} if $$k2^{n}-1$$ is composite for every positive $$n$$. On the other hand, a positive integer $$n$$ is a noncotient provided $$n\neq m-\varphi(m)$$. This paper is devoted to prove two main results. In the first one, the authors provide a list of ten Riesel numbers $$k$$ such that $$k2^n$$ is noncototient for every $$n$$. The second main result goes in the opposite direction, and is in fact more interesting. It states that there are infinitely many Riesel numbers $$k$$ with the property that $$k2^n$$ is a cototient for some $$n$$. The proofs are of elementary nature and rely on some Mathematica computations. Reviewer: Antonio M. Oller Marcén (Zaragoza) On the characteristic polynomial of $$(k,p)$$-Fibonacci sequence https://zbmath.org/1485.11038 2022-06-24T15:10:38.853281Z "Trojovský, Pavel" https://zbmath.org/authors/?q=ai:trojovsky.pavel Summary: Recently, Bednarz introduced a new two-parameter generalization of the Fibonacci sequence, which is called the $$(k,p)$$-Fibonacci sequence and denoted by $$(F_{k,p}(n))_{n\geq0}$$. In this paper, we study the geometry of roots of the characteristic polynomial of this sequence. Note on a theorem of Zehnxiag Zhang https://zbmath.org/1485.11054 2022-06-24T15:10:38.853281Z "Laib, I." https://zbmath.org/authors/?q=ai:laib.ilias "Derbal, A." https://zbmath.org/authors/?q=ai:derbal.abdellah "Mechik, R." https://zbmath.org/authors/?q=ai:mechik.rachid "Rezzoug, N." https://zbmath.org/authors/?q=ai:rezzoug.noreddine Summary: A sequence of strictly positive integers is said to be primitive if none of its terms divides the others. In this paper, we give a new proof of a result, conjectured by P. Erdős and Z. Zhang in 1993 [Proc. Am. Math. Soc. 117, No. 4, 891--895 (1993; Zbl 0776.11013)], on a primitive sequence whose the number of the prime factors of the terms counted with multiplicity is at most 4. The objective of this proof is to improve the complexity, which helps to prove this conjecture. Primes in Beatty sequence https://zbmath.org/1485.11059 2022-06-24T15:10:38.853281Z "Karthick Babu, C. G." https://zbmath.org/authors/?q=ai:karthick-babu.c-g Summary: For a polynomial $$g(x)$$ of $$\deg k\ge 2$$ with integer coefficients and positive integer leading coefficient, we prove an upper bound for the least prime $$p$$ such that $$g(p)$$ is in non-homogeneous Beatty sequence $$\{\lfloor\alpha n+\beta\rfloor:n=1,2,3,\dots\}$$, where $$\alpha,\beta\in\mathbb{R}$$ with $$\alpha>1$$ is irrational and we prove an asymptotic formula for the number of primes $$p$$ such that $$g(p)=\lfloor\alpha n+\beta\rfloor$$. Next, we obtain an asymptotic formula for the number of primes $$p$$ of the form $$p=\lfloor\alpha n+\beta\rfloor$$ which also satisfies $$p\equiv f\pmod d$$, where $$f,d$$ are integers with $$1\leq f<d$$ and $$(f,d)=1$$. The flatness of ternary cyclotomic polynomials https://zbmath.org/1485.11067 2022-06-24T15:10:38.853281Z "Zhang, Bin" https://zbmath.org/authors/?q=ai:zhang.bin.3|zhang.bin.4|zhang.bin.1|zhang.bin.2 Summary: It is well known that all of the prime cyclotomic polynomials and binary cyclotomic polynomials are flat, and the flatness of ternary cyclotomic polynomials is much more complicated. Let $$p<q<r$$ be odd primes such that $$zr\equiv\pm\pmod{pq}$$, where $$z$$ is a positive integer. So far, the classification of flat ternary cyclotomic polynomials for $$1\leq z\leq 5$$ has been given. In this paper, for $$z=6$$ and $$q\equiv\pm 1\pmod{p}$$, we give the necessary and sufficient conditions for ternary cyclotomic polynomials $$\Phi_{pqr}(x)$$ to be flat. Sign changes in the prime number theorem https://zbmath.org/1485.11130 2022-06-24T15:10:38.853281Z "Morrill, Thomas" https://zbmath.org/authors/?q=ai:morrill.thomas "Platt, Dave" https://zbmath.org/authors/?q=ai:platt.dave "Trudgian, Tim" https://zbmath.org/authors/?q=ai:trudgian.tim Let $$V(T)$$ denote the number of sign changes in $$\psi(x)-x$$ for $$x\in[1,T]$$, where $$\psi(x)=\sum_{p^m\leq x}\log p$$. It is known that $\liminf_{T\to\infty}\frac{V(T)}{\log T}\geq\frac{\gamma^\ast}{\pi},$ where $$\gamma^\ast$$ defined as follows. Let $$\Theta$$ denote the supremum over $$\Re(\rho)$$ where $$\rho$$ ranges over all zeroes of the Riemann zeta function $$\zeta(s)$$. If there are any zeroes $$\rho=\beta +i\gamma$$ with $$\beta=\Theta$$ we shall define $$\gamma^\ast$$ as the least positive $$\gamma$$; otherwise $$\gamma^\ast=\infty$$. Thus, if the Riemann hypothesis is true, then $$\gamma^\ast=\gamma_1$$, where $$\gamma_1$$ is the imaginary part of the lowest-lying non-trivial zero of $$\zeta(s)$$. If the Riemann hypothesis is false then, by a recent to appear work of the second and third authors, $$\gamma^\ast\geq 3\,000\,175\,332\,800>3\times 10^{12}$$. \textit{J. Kaczorowski} [Acta Arith. 59, No. 1, 37--58 (1991; Zbl 0736.11048)] proved unconditionally that $\liminf_{T\to\infty}\frac{V(T)}{\log T}\geq\frac{\gamma_1}{\pi}+10^{-250}.$ In this paper, the authors follow Kaczorowski's method, making some theoretical and computational improvements, and prove the above with $$1.867\times 10^{-30}$$ replacing $$10^{-250}$$. Reviewer: Mehdi Hassani (Zanjan) A new elementary proof of the Prime Number Theorem https://zbmath.org/1485.11139 2022-06-24T15:10:38.853281Z "Richter, Florian K." https://zbmath.org/authors/?q=ai:richter.florian-karl Elementary proofs of the Prime Number Theorem (PNT), does not necessarily mean simple, but refers to methods that avoid using complex analysis and instead rely only on rudimentary facts from calculus and basic arithmetic identities and inequalities. Although it was believed for a long time not to be possible, such a proof was eventually found by Erdős and Selberg, independently, both based on a Selberg's fundamental formula''. Later, other elementary ways of proving PNT were found by Daboussi using what he called the convolution method'' and by Hildebrand, which relies on a corollary of the large sieve. The purpose of the paper under review is to provide yet another elementary proof of PNT, running over the comparing the mean values of the combined arithmetic functions $$f\left(\Omega(n)+1\right)$$ and $$f\left(\Omega(n)\right)$$, where $$\Omega(n)$$ denote the number of prime factors of a positive integer $$n$$ (counted with multiplicities), and $$f:\mathbb{N}\to\mathbb{C}$$ is any bounded arithmetic function. More precisely, the author proves that, as $$N\to\infty$$, one has $\frac{1}{N}\sum_{n=1}^N f\left(\Omega(n)+1\right)=\frac{1}{N}\sum_{n=1}^N f\left(\Omega(n)\right)+o(1).$ If we let $$f(n)=(-1)^n$$ then $$f\left(\Omega(n)\right)=\lambda(n)=\left(-1\right)^{\Omega(n)}$$ coincides with the classical Liouville function $$\lambda(n)$$, and from the above identity we deduce that $$\sum_{n=1}^N\lambda(n)=o(N)$$, which is a well-known equivalent form of PNT. It is worth noting that this is the first proof of PNT that builds on Chebyshev's original idea of estimating the number of primes in the interval $$(n,2n]$$. Reviewer: Mehdi Hassani (Zanjan) Random multiplicative functions: the Selberg-Delange class https://zbmath.org/1485.11140 2022-06-24T15:10:38.853281Z "Aymone, Marco" https://zbmath.org/authors/?q=ai:aymone.marco Let $$\beta\in[1/2,1)$$, $$p$$ be a generic prime number and $$f_\beta$$ be a random multiplicative function supported on the squarefree integers such that $$(f_\beta(p))_p$$ is an i.i.d. sequence of random variables with distribution $$\mathbb{P}(f(p)=-1)=\beta=1-\mathbb{P}(f(p)=+1)$$. In the paper under review, the author is motivated by the behaviour of partial sums of Möbius function and its relation with the Riemann hypothesis, more precisely Wintner's model in this area. The author considers the following questions: Question 1. What can be said about the partial sums $$\sum_{n\leq x}f_\beta(n)$$ for $$\beta\in(1/2,1)$$? Do they have square root cancellation as in Wintner's model and as we expect for the Möbius function under the Riemann hypothesis? Question 2. If the partial sums $$\sum_{n\leq x}f_\beta(n)$$ are $$O(x^{1-\delta})$$ for some $$\delta>0$$, almost surely, then can we say something about the partial sums of the Möbius function? The author gives a negative answer for Question 1 by showing that for any integer $$n\geq 1$$ and $$\beta=1-1/2^{n+1}$$, for any $$\delta>0$$, $$\sum_{n\leq x}f_\beta(n)$$ is not $$O(x^{1-\delta})$$ almost surely. Regarding to Question 2, letting $$\omega(n)$$ to be be the number of distinct primes that divide $$n$$, he proves that the Riemann hypothesis is equivalent to the following statement: $\sum_{n\leq x}(2\beta-1)^{-\omega(n)}f_\beta(n)=O(x^{1/2+\varepsilon}),$ for all $$\varepsilon>0$$ and $$x$$ sufficiently large with respect to $$\varepsilon$$, almost surely, for each $$\beta\in(1/2+1/(2\sqrt{2}),1)$$. Hence, the Riemann hypothesis is equivalent to the square root cancellation of the above weighted partial sums of $$f_\beta$$. Reviewer: Mehdi Hassani (Zanjan) On factorizations into coprime parts https://zbmath.org/1485.11141 2022-06-24T15:10:38.853281Z "Just, Matthew" https://zbmath.org/authors/?q=ai:just.matthew "Lebowitz-Lockard, Noah" https://zbmath.org/authors/?q=ai:lebowitz-lockard.noah Let $$f(n)$$ and $$g(n)$$ be the number of unordered and ordered factorizations of $$n$$ into integers larger than $$1$$. Moments $$\sum_{n\leq x}f(n)^\beta$$ and $$\sum_{n\leq x}g(n)^\beta$$ for $$\beta>0$$ are known in literature. In this paper, the authors considers the arithmetic functions $$F(n)$$ and $$G(n)$$, counting the number of unordered and ordered factorizations of $$n$$ into pairwise coprime integers larger than one, and estimate the moments $$\sum_{n\leq x}F(n)^\beta$$ and $$\sum_{n\leq x}G(n)^\beta$$ for all real values of $$\beta$$. More precisely, for the case $$\beta=1$$ they prove that $\sum_{n\leq x}F(n)=x\,e^{(2\sqrt{2}e^{-\gamma/2}+o(1))(\log x)^{1/2}(\log\log x)^{-1/2}},$ and $\sum_{n\leq x}G(n)=x\,e^{(c+o(1))(\log x)(\log\log x)^{-1}},$ where $$\gamma$$ is the Euler-Mascheroni constant, and the constant $$c$$ is computable constant in terms of the specific values of Lambert $$W$$ function and exponential integral function. Meanwhile, the authors estimate the moments $$\sum_{n\leq x}f(n)^\beta$$ and $$\sum_{n\leq x}g(n)^\beta$$ for $$\beta<0$$. Reviewer: Mehdi Hassani (Zanjan) The distribution of numbers with many factorizations https://zbmath.org/1485.11142 2022-06-24T15:10:38.853281Z "Pollack, Paul" https://zbmath.org/authors/?q=ai:pollack.paul In the paper under review, the author studies the arithmetic function $$f(n)$$ counting the number of ways of writing $$n$$ as product of integers $$\geqslant 2$$, where the order of the factors is not taken into account. This function was first introduced by \textit{P. A. MacMahon} [Proc. Lond. Math. Soc. (2) 22, 404--411 (1924; JFM 50.0083.02)], and its average order was discovered by \textit{A. Oppenheim} [J. Lond. Math. Soc. 2, 123--130 (1927; JFM 53.0157.02)]. In this interesting work, the author first extends a previous result of \textit{E. R. Canfield} et al. [J. Number Theory 17, 1--28 (1983; Zbl 0513.10043)] by showing that, for fixed $$\varepsilon > 0$$ and $$\alpha \in \left( 0,1 \right)$$, there exists a real number $$x_0 = x_0(\varepsilon,\alpha) > 0$$ such that, for all $$x > x_0$$ and each subset $$S$$ of $$\left[ 1,x \right] \cap \mathbb{Z}$$ of cardinality $$|S| \leqslant x^{1-\alpha}$$, we have $\sum_{n \in S} f(n) \leqslant x \, \exp \left( (\varepsilon - \alpha) \frac{\log x \log_3 x}{\log_2 x}\right)$ where $$\log_k$$ is the $$k$$-fold iterated logarithm. As a consequence, the author establishes an estimate for the moments $$\sum_{n \leqslant x} f(n)^\beta$$ with $$\beta > 1$$. Finally, the author investigates the more intricate case when $$0 < \beta < 1$$. Using a completely different method, he proves that, when $$\beta \in \left( 0,1 \right)$$ is fixed, then, for all $$x$$ sufficiently large, the estimate $\sum_{n \leqslant x} f(n)^\beta = x \, \exp \left( (1+o(1)) (1-\beta)^{\frac{1}{1-\beta}} \log_2 x \left( \frac{\log_2 x}{\log_3 x}\right)^{\frac{\beta}{1-\beta}}\right).$ holds. The lower bound uses the fact that, when $$n$$ is squarefree with $$\omega(n) = k$$, then $$f(n) = B_k$$ the $$k$$th Bell number, so that $$\sum_{n \leqslant x} f(n)^\beta \geqslant B_k^\beta \pi_k (x)$$, whereas the upper bound follows from an observation of Oppenheim along with some effective uniform bounds for powers of the Dirichlet-Piltz divisor $$\tau_z(n)$$ proved by \textit{K. K. Norton} [J. Number Theory 40, No. 1, 60--85 (1992; Zbl 0748.11046)]. Reviewer: Olivier Bordellès (Aiguilhe) On Euclidean ideal classes in certain abelian extensions https://zbmath.org/1485.11154 2022-06-24T15:10:38.853281Z "Deshouillers, J.-M." https://zbmath.org/authors/?q=ai:deshouillers.jean-marc "Gun, S." https://zbmath.org/authors/?q=ai:gun.sanoli "Sivaraman, J." https://zbmath.org/authors/?q=ai:sivaraman.jyothsnaa For an algebraic number field $$K$$, let $${\mathcal O}_K$$, $${\mathcal O}_K^\times$$, and $${\text {Cl}}_K$$ be, respectively, its ring of integers, unit group, and class group. Let $$E_K$$ be the set of all fractional ideals of $$K$$ containing $${\mathcal O}_K$$. Assume $${\mathcal O}_K^\times$$ is infinite and call its rank the unit rank of $$K$$. An ideal class $$[\mathfrak J]$$ of $${\text {Cl}}_K$$ is called a Euclidean ideal class if there exists a map $$\psi:E_K\rightarrow {\mathbb N}$$ such that for any ideal $$\mathfrak a \in [\mathfrak J]$$ and for all ideals $$\mathfrak b \in E_K$$ and for all $$x\in \mathfrak {ab}\setminus \mathfrak a$$, there exists $$z\in x+\mathfrak a$$ such that $\psi(z^{-1}\mathfrak {ab})<\psi(\mathfrak b).$ By restricting to a finer family of number fields, the authors of the present paper are able to improve upon a result obtained in [\textit{H. Graves} and \textit{M. R. Murty}, Proc. Am. Math. Soc. 141, No. 9, 2979--2990 (2013; Zbl 1329.11115)] by proving the following theorem. Theorem 1. Suppose that $$K$$ is a number field with unit rank greater than or equal to 3 and its Hilbert class field $$H(K)$$ is abelian over $$\mathbb Q$$. Also suppose that the conductor of $$K$$ is $$f$$ and $${\mathbb Q}(\zeta_f)$$ over $$K$$ is cyclic. Then $${\text {Cl}}_K$$ is cyclic if and only if it has a Euclidean ideal class. As an immediate corollary they obtain Corollary 2. Let $$K$$ be an abelian number field with conductor $$f$$ and assume the unit rank of $$K$$ is greater than or equal to 3. If $${\mathbb Q}(\zeta_f)$$ over $$K$$ is cyclic, then $${\mathcal O}_K$$ is Euclidean if and only if it is a PID. Under the assumption that the \textit{P. D. T. A. Elliott} and \textit{H. Halberstam} [in: Sympos. Math., Roma 4, Teoria numeri Dic. 1968, e Algebra, Marzo 1969, 59--72 (1970; Zbl 0238.10030)] conjecture is true, the authors are able to strengthen the above results. They prove Theorem 3. Let $$K$$ be a number field such that the Hilbert class field $$H(K)$$ is abelian and the Galois group $$\text{Gal}({\mathbb Q}(\zeta_f)/K)$$ is cyclic where $$f$$ is the conductor of $$K$$. Now if the Elliott and Halberstam conjecture is true and the unit rank of $$K$$ is strictly greater than one, then $${\text {Cl}}_K$$ is cyclic if and only if it has a Euclidean ideal class. As an immediate corollary they obtain Corollary 4. Let $$K$$ be an abelian number field with conductor $$f$$ and assume the unit rank of $$K$$ is strictly greater than one. Suppose that the Elliott and Halberstam conjecture is true. If $${\mathbb Q}(\zeta_f)$$ over $$K$$ is cyclic, then $${\mathcal O}_K$$ is Euclidean if and only if it is a PID. The proofs of Theorems 1 and 3 assume that $$f$$ is the smallest even integer such that $$K\subseteq {\mathbb Q}(\zeta_f)$$. However, in the final section of the paper the authors show that their arguments used to prove Theorems 1 and 3 actually prove stronger results in the sense that these latter results include number fields $$K\subset {\mathbb Q}(\zeta_{2^k})$$, for all $$k\geq1$$ where $${\mathcal O}_K$$ is a PID and $$\zeta_{2^k}$$ is the $$2^k$$-th primitive root of unity. Reviewer: James E. Carter (Charleston) A density of ramified primes https://zbmath.org/1485.11159 2022-06-24T15:10:38.853281Z "Chan, Stephanie" https://zbmath.org/authors/?q=ai:chan.stephanie "McMeekin, Christine" https://zbmath.org/authors/?q=ai:mcmeekin.christine "Milovic, Djordjo" https://zbmath.org/authors/?q=ai:milovic.djordjo-z \textit{J. B. Friedlander} et al. [Invent. Math. 193, No. 3, 697--749 (2013; Zbl 1296.11150)] considered totally real cyclic extensions $$K/\mathbb Q$$ in which every totally positive integer is a square and showed that if $$\sigma$$ denotes a generator of $$\operatorname{Gal}(K/\mathbb Q)$$, then the set of principal prime ideals $$\pi \mathbb Z_K$$ which split in $$K(\sqrt{\sigma(\pi)})$$ is of density $$1/2$$. A certain generalization has been later provided by \textit{P. Koymans} and \textit{D. Milovic} [Duke Math. J. 170, No. 8, 1723--1755 (2021; Zbl 1478.11109)]. The authors deal with totally real Galois extensions $$K/\mathbb Q$$ of odd degree $$n$$ with odd narrow class number in which $$2$$ is inert. Let $$p$$ be a prime splitting in $$K$$ and for prime ideals $$\mathfrak p$$ lying over $$p$$ denote by $$R_\mathfrak p^+$$ the maximal abelian extension of $$K$$ unramified at all prime ideals $$\ne \mathfrak p$$. Let $$K_\mathfrak p$$ be the unique quadratic extension of $$K$$ lying in $$R_\mathfrak p^+$$ and let $$K(p)$$ be the composite of all fields $$K_\mathfrak p$$. For $$\mu\in\{-1,1\}$$ denote by $$S_\mu$$ the set of all primes $$p\equiv\mu$$ mod $$4$$, splitting in $$K$$ and by $$F_\mu$$ the subset of $$S_\mu$$ consisting of primes whose prime divisors in $$K(p)$$ are of the first degree. Denote by $$F_\mu(x),S_{\mu}(x)$$ the counting functions of the sets $$F_\mu$$, resp. $$S_{\mu}$$. The main result (Theorem 1) shows that if for all real characters $$\chi\ne \chi_0=1$$ of conductors $$q\le Q$$ and all integers $$M$$ and $$N\le Q^\eta$$ with $$\eta=2/n(n-1)$$ one has for every $$\varepsilon>0$$ $\left|\sum_{a=M+1}^N\chi(n)\right|\le cQ^{\eta(1-\delta)+\varepsilon}$ with some $$\delta>0$$ and $$c=c(\varepsilon)$$ (this is a variant of the conjecture stated by Friedlander et al. [loc. cit.]), then there is an explicit formula for the value of the limit $$\lim_{x\to\infty}F_\mu(x)/S_\mu(x)$$, which is positive. In the case $$n=3$$ this result is shown unconditionally in Theorem 2, the limit being equal to $$1/8$$ for $$\mu=1$$ and to $$3/8$$ for $$\mu=-1$$. Reviewer: Władysław Narkiewicz (Wrocław) Strongly obtuse rational lattice triangles https://zbmath.org/1485.37033 2022-06-24T15:10:38.853281Z "Larsen, Anne" https://zbmath.org/authors/?q=ai:larsen.anne "Norton, Chaya" https://zbmath.org/authors/?q=ai:norton.chaya "Zykoski, Bradley" https://zbmath.org/authors/?q=ai:zykoski.bradley \textit{A. N. Zemlyakov} and \textit{A. B. Katok} [Mat. Zametki 18, 291--300 (1975; Zbl 0315.58014)] used the method of unfolding'' to transform a piecewise linear billiard path in a polygonal table whose angles are rational multiples of $$\pi$$ into a straight path on a translation surface obtained by unfolding the polygon. \textit{W. A. Veech} [Invent. Math. 97, No. 3, 553--583 (1989; Zbl 0676.32006)] showed that a translation surface whose affine automorphism group is a lattice has the property that a straight line flow in a given direction is either periodic or uniquely ergodic, and showed that the unfolding of an obtuse isosceles triangle with angles $$(\frac{\pi}{n},\frac{\pi}{n},\frac{(n-2)\pi}{n})$$ has this property. \textit{R. Kenyon} and \textit{J. Smillie} [Comment. Math. Helv. 75, No. 1, 65--108 (2000; Zbl 0967.37019)] formulated a number-theoretic criterion for the angles of an acute rational triangle necessary for the lattice property, and used this to classify all acute and right triangles (the argument relying on a computer search, initially leaving open the possibility of further cases not found, a possibility later eliminated by work of \textit{J.-C. Puchta} [Comment. Math. Helv. 76, No. 3, 501--505 (2001; Zbl 1192.37048)]). Here a contribution is made to the obtuse case, with a classification of triangles which unfold to Veech surfaces when the largest angle is at least $$\frac{3\pi}{4}$$. In the case of largest angle greater than $$\frac{2\pi}{3}$$, it is shown that the unfolding does not have the Veech property except possibly if it belongs to one of six identified infinite families. The approach uses a criterion of \textit{M. Mirzakhani} and \textit{A. Wright} [Duke Math. J. 167, No. 1, 1--40 (2018; Zbl 1435.32016)] itself developed from work of \textit{M. Möller} [J. Am. Math. Soc. 19, No. 2, 327--344 (2006; Zbl 1090.32004)] and \textit{C. T. McMullen} [J. Am. Math. Soc. 16, No. 4, 857--885 (2003; Zbl 1030.32012)]. Reviewer: Thomas B. Ward (Newcastle)