Recent zbMATH articles in MSC 11Mhttps://zbmath.org/atom/cc/11M2022-09-13T20:28:31.338867ZWerkzeugHolomorphy 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)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.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)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)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.