Recent zbMATH articles in MSC 11Mhttps://www.zbmath.org/atom/cc/11M2022-01-14T13:23:02.489162ZWerkzeugSigns of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic formhttps://www.zbmath.org/1475.110802022-01-14T13:23:02.489162Z"Vaishya, Lalit"https://www.zbmath.org/authors/?q=ai:vaishya.lalitSummary: In this article, we establish that there are infinitely many sign changes of Fourier coefficients of a normalised Hecke eigenform supported at positive integers represented by a primitive integral binary quadratic form with negative discriminant whose class number is 1. We also provide a quantitative result for the number of such sign changes in the interval \((x, 2x]\) for sufficiently large \(x\).A note on Burgess boundhttps://www.zbmath.org/1475.110912022-01-14T13:23:02.489162Z"Munshi, Ritabrata"https://www.zbmath.org/authors/?q=ai:munshi.ritabrataSummary: Let \(f\) be a \(\mathrm{SL}(2,\mathbb{Z})\) Hecke cusp form, and let \(\chi\) be a primitive Dirichlet character modulo \(M\), which we assume to be prime. We prove the Burgess-type bound for the twisted \(L\)-function:
\[
\begin{aligned} L\left( \tfrac{1}{2},f\otimes \chi \right) \ll_{f,\varepsilon } M^{1/2-1/8+\varepsilon}.\end{aligned}
\]
The method also yields the original bound of Burgess for Dirichlet L-functions:
\[
\begin{aligned}L\left(\frac{1}{2},\chi\right)\ll_{\varepsilon} M^{1/4-1/16+\varepsilon}.\end{aligned}
\]
For the entire collection see [Zbl 1403.11002].Prime geodesic theorem for the Picard manifoldhttps://www.zbmath.org/1475.111022022-01-14T13:23:02.489162Z"Balkanova, Olga"https://www.zbmath.org/authors/?q=ai:balkanova.olga-g"Frolenkov, Dmitry"https://www.zbmath.org/authors/?q=ai:frolenkov.dmitrii-aLet \(\mathbb{H}^3\) be the 3-\(d\) hyperbolic Riemannian space, \(\mathrm{PSL}(2,\mathbb{C})\simeq \mathrm{Isom}(\mathbb{H}^3)\) its isometry group and \(\Gamma\) a discrete cofinite torsion free group in \(\mathrm{Isom}(\mathbb{H}^3)\). Any non trivial closed geodesic is associated uniquely to a conjugacy class \([\gamma]\) of an hyperbolic or loxodromic group element whose norm \(N(\gamma)\) determines its length. The function \(\pi_\Gamma(X)=\{[\gamma]|N(\gamma)\le X\}\) counts the closed geodesic on \(X_\Gamma=\Gamma \backslash\mathbb{H}^3\) according to their length. The counting function \(\pi_\Gamma(X)\) decomposes along \(\pi_\Gamma(X)=\mathrm{Li}(X^2)+E_\Gamma(X)\) as the sum of a principal asymptotic term and a remainder: upper bounds on this remainder \(E_\Gamma(X)\), of great interest in number theory, are called Prime Geodesic Theorem (PGT), giving finer and finer error estimates with different asymptotics. \textit{P. Sarnak} [Acta Math. 151, 253--295 (1983; Zbl 0527.10022)] established the upper bound \(E_\Gamma(X)=\mathcal{O}(X^{5/3+\varepsilon})\) with any small \(\varepsilon>0\). In case of the Picard group \(\Gamma=\mathrm{PSL}(2,\mathbb Z[\mathrm{i}])\) induced by the Gaussian integers \(\mathbb Z[\mathrm{i}]\), different PGT have been proved, either with a \(L\)-function hypothesis by \textit{S. Y. Koyama} [Forum Math. 13, No. 6, 781--793 (2001; Zbl 1061.11024)] or unconditionally by \textit{O. Balkanova} et al. [Trans. Am. Math. Soc. 372, No. 8, 5355--5374 (2019; Zbl 07121875)]. The main result proved in this paper is the PGT for the Picard manifold
\[
E_\Gamma(X)=\mathcal{O}\left(X^{3/2+\theta/2+\varepsilon}\right).
\]
Here \(\varepsilon>0\) is a priori given, while \(\theta\) denotes a subconvexity exponent for quadratic Dirichlet \(L\)-function defined over Gaussian integers: \textit{P. Nelson} [``Eisenstein series and the cubic moment for \(\mathrm{PGL}_2\)'', Preprint, \url{arXiv.1911.06310}] proved that \(\theta=1/3\) may be taken. The proof of this new remainder estimate is based on two upper bounds
\begin{itemize}
\item For \(X\) big enough, \(T\in [X^\varepsilon,X^{1/2}]\) and \(|t|=\mathcal O(T^\varepsilon)\)
\[
\sum_{r_j}\frac{r_j}{\sinh(\pi r_j)}\omega_\Gamma(r_j) X^{\mathrm{i}r_j}L(u_j\otimes u_j, 1/2+it)=\mathcal{O}\left(T^{3/2}X^{1/2+\theta+\varepsilon}\right).\tag{1}
\]
\item For \(T\in[1,X^{1/2}]\),
\[
\sum_{0<r_j\le T}X^{\mathrm{i}r_j}=\mathcal{O}\left(X^{(1+\theta)/2}T(TX)^\varepsilon\right).\tag{2}
\]
\end{itemize}
Here the family \((u_j,1+r_j^2)\) is a maximal orthonormal cusp forms \((u_j)\) basis, with corresponding eigenvalues \(1+r_j^2\), the function sums \(\omega_T\) is a smooth characteristic function of the interval \([T,2T]\) and \(L(u\otimes u,s)\) is a Maaß-Rankin-Selberg \(L\)-function.
The proof core is to establish the upper bound (1) which implies the estimate (2), the PGS being a consequence of (2). The main idea to prove (1) is to avoid the usual way where bounds are obtained through crude absolute value estimates on terms in the (1) left-hand side. Instead the authors introduce exact formulæ for the first moment of Maaß-Rankin-Selberg which allows to consider oscillations of the exponentials \(X^{\mathrm{i}r_j}\). This method has been used successfully for the 2-d modular surface \(\mathrm{PSL}(2,\mathbb{Z})\backslash\mathbb{H}^2\) by the two authors in [J. Lond. Math. Soc., II. Ser. 99, No. 2, 249--272 (2019; Zbl 1456.11092)] with improvements on already known PGTs. The current proof used Kuznetsov formula to win finer control on Kloosterman sums. Some features in 3-d are new (e.g. geometry of the Picard manifold, special functions in the trace formula) and are solved in the particular case of the Picard manifold.
Reviewer: Laurent Guillopé (Nantes)A note on additive twists, reciprocity laws and quantum modular formshttps://www.zbmath.org/1475.111112022-01-14T13:23:02.489162Z"Nordentoft, Asbjørn Christian"https://www.zbmath.org/authors/?q=ai:nordentoft.asbjorn-christianSummary: We prove that the central values of additive twists of a cuspidal \(L\)-function define a quantum modular form in the sense of Zagier, generalizing recent results of \textit{S. Bettin} and \textit{S. Drappeau} [``Limit laws for rational continued fractions and value distribution of quantum modular forms'', Preprint, \url{arXiv:1903.00457}]. From this, we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal \(L\)-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet \(L\)-functions. Furthermore, we give an interpretation of quantum modularity at infinity for additive twists of \(L\)-functions of weight 2 cusp forms in terms of the corresponding functional equations.Effective universality theorem: a surveyhttps://www.zbmath.org/1475.111462022-01-14T13:23:02.489162Z"Garunkštis, Ramūnas"https://www.zbmath.org/authors/?q=ai:garunkstis.ramunas"Laurinčikas, Antanas"https://www.zbmath.org/authors/?q=ai:laurincikas.antanasSummary: In 1975, S. M. Voronin proved the universality theorem for the Riemann zeta-function. This famous theorem is ineffective. Here we survey results related to the effectivization of Voronin's theorem.On the mean value of generalized Dirichlet \(L\)-functions with weight of the character sumshttps://www.zbmath.org/1475.111472022-01-14T13:23:02.489162Z"Ma, Rong"https://www.zbmath.org/authors/?q=ai:ma.rong"Niu, Yana"https://www.zbmath.org/authors/?q=ai:niu.yana"Wang, Haodong"https://www.zbmath.org/authors/?q=ai:wang.haodong"Zhang, Yulong"https://www.zbmath.org/authors/?q=ai:zhang.yulongSummary: Let \(p\) be a prime, \(\chi\) denote a Dirichlet character modulo \(p\). For any integer \(x (1\leq x\leq p-1), \bar{x}\) denotes the integer inverse of \(x\) such that \(x\bar{x}\equiv 1\pmod{p}\), we study the following mean value of a kind of character sums with generalized Dirichlet \(L\)-functions
\[
\sum\limits_{\substack{\chi (-1)=1 \\ \chi \neq \chi_0}} \left| \sum\limits_{x=1}^{p-1} \chi (x+\bar{x})\right|^2 |L(1,\chi ,a)|^2,
\]
where \(\chi_0\) is the principal character modulo \(p\), and \(L(1,\chi ,a)\) is the generalized Dirichlet \(L\)-functions. In this paper, we will use the analytic method and get a sharp asymptotic formula.Counting zeros of Dirichlet \(L\)-functionshttps://www.zbmath.org/1475.111482022-01-14T13:23:02.489162Z"Bennett, Michael A."https://www.zbmath.org/authors/?q=ai:bennett.michael-a"Martin, Greg"https://www.zbmath.org/authors/?q=ai:martin.greg"O'Bryant, Kevin"https://www.zbmath.org/authors/?q=ai:obryant.kevin"Rechnitzer, Andrew"https://www.zbmath.org/authors/?q=ai:rechnitzer.andrew-danielSummary: We give explicit upper and lower bounds for \(N(T,\chi)\), the number of zeros of a Dirichlet \(L\)-function with character \(\chi\) and height at most \(T\). Suppose that \(\chi\) has conductor \(q> 1\), and that \(T\geq 5/7\). If \(\ell=\log \frac{q(T+2)}{2\pi}> 1.567\), then
\[
\left\vert N(T,\chi)-\left(\frac{T}{\pi}\log\frac{qT}{2\pi e}\frac{\chi(-1)}{4}\right)\right\vert\le 0.22737\ell+2\log(1+\ell)-0.5.
\]
We give slightly stronger results for small \(q\) and \(T\). Along the way, we prove a new bound on \(\vert L(s,\chi)\vert\) for \(\sigma< -1/2\).Alternating multiple zeta values, and explicit formulas of some Euler-Apéry-type serieshttps://www.zbmath.org/1475.111492022-01-14T13:23:02.489162Z"Wang, Weiping"https://www.zbmath.org/authors/?q=ai:wang.weiping|wang.weiping.1"Xu, Ce"https://www.zbmath.org/authors/?q=ai:xu.ceSummary: In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to \(\ln(2)\), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma function, polygamma function and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as examples.On the Poincaré expansion of the Hurwitz zeta functionhttps://www.zbmath.org/1475.111502022-01-14T13:23:02.489162Z"Fejzullahu, Bujar"https://www.zbmath.org/authors/?q=ai:fejzullahu.bujar-xhSummary: In this paper, we extend the result of \textit{R. B. Paris} [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2053, 297--304 (2005; Zbl 1145.11326)] on the exponentially improved expansion of the Hurwitz zeta function \(\zeta (s, z)\), the expansion of which can be reduced to the large-\(z\) Poincaré asymptotics of \(\zeta (s, z)\). Furthermore, we deduce some new series and integral representations of the Hurwitz zeta function \(\zeta (s, z)\).Fractional calculus, zeta functions and Shannon entropyhttps://www.zbmath.org/1475.111512022-01-14T13:23:02.489162Z"Guariglia, Emanuel"https://www.zbmath.org/authors/?q=ai:guariglia.emanuelSummary: This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz \(\zeta\) function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.Diagonal convergence of the remainder Padé approximants for the Hurwitz zeta functionhttps://www.zbmath.org/1475.111522022-01-14T13:23:02.489162Z"Prévost, M."https://www.zbmath.org/authors/?q=ai:prevost.marc"Rivoal, T."https://www.zbmath.org/authors/?q=ai:rivoal.tanguyThis interesting paper constructs sequences of complex numbers that rapidly converge to the Hurwitz zeta function
\[
\zeta(s,a)=\sum_{k=0}^{\infty}\,\frac{1}{(k+a)^s},\ \Re(a)>0\text{ and }\Re(s)>1.
\]
The method uses diagonal Padé approximants to the remainder series \[\sum_{k=n}^{\infty}\,\frac{1}{(k+a)^s}.\] An important tool is is the series
\[
\Phi_s(z)=\sum_{n=0}^{\infty}\,\frac{(s)_{2n+1}}{(2n+2)!}B_{2n+2}(-z)^n\, z\not=0\text{ and }s>0,
\]
with \((B_{2n+2})_{n\geq 0}\) the Bernoulli numbers, which is the asymptotic expansion of
\[
\hat{\Phi}_s(z)=\int_0^{\infty}\,\frac{\mu_s(x)}{1-zx}\,dx,
\]
with \(\mu_s(x)=\omega_s(\sqrt{x})/2\sqrt{x}\in L^1(\mathbb{R}^{+}\) for \(\Re(s)>0\) and
\[
\omega_s(x)=\frac{2(-1)^mx^s}{\Gamma(s)\Gamma(m+1-s)}\,\int_x^{\infty}\,(t-x)^{m-s}\frac{d^m}{dt^m}\left(\frac{1}{e^{2\pi t}-1}\right)\,dt.
\]
The main results are:
\textbf{Theorem 1.} (\S1) Let \(s>0,\,s\not= 1\) and \(a\in\mathbb{C}\) such that \(\Re(a)>0\). Set \(a_n=n+a\). Then, for every large enough integer \(n\) and any integer \(k\geq 1\), we have
\[
\zeta(s,a)=\sum_{j=0}^{n-1}\frac{1}{(j+a)^s}+\frac{1}{(s-1)a_n^{s-1}}+\frac{1}{2a_n^s}+\frac{1}{a_n^{s+1}}[k/k]_{\Phi_s}\left(-\frac{1}{a_n^2}\right)+ \varepsilon_{k,s} \left(\frac{1}{a_n^2}\right),
\]
where
\[
| \varepsilon_{k,s} (1/a_n^2)|\leq D_s\frac{(2k+2\rho)\Gamma(2k+\rho+1)^2}{|a_n|^{4k+2}(4k+2\rho+1)(2k+1)\left(\begin{matrix}4k+2\rho\\ 2k+1\end{matrix}\right)^2},
\]
where \(\rho=(m+7)/2\) and \(D_s=(2\pi)^sm!/\gamma(s)\) and \(m=[s]\).
\textbf{Corollary 1.} (\S1) Let \(r\in\mathbb{Q}\) such that \(0<r<2e\). Let \(s>0,s\not= 1\). Then, for every integer \(n\geq 1\) such that \(rn\) is an integer, we have
\[
\zeta(s)=\sum_{k=1}^n\frac{1}{k^s}+\frac{1}{(s-1)n^{s-1}}-\frac{1}{2n^s}+\frac{1}{n^{s+1}}[rn/rn]-{\Phi_s}\left(-\frac{1}{n^2}\right)+\delta_{r,s,n},
\]
where
\[
\limsup_{n\rightarrow\infty}\,|\delta_{r,s,n}|^{1/n}\leq\left(\frac{r}{2e}\right)^{4r}.
\]
\textbf{Proposition 1.} (\S3) For any \(s>0\) and any \(x\geq 0\), we have
\[
0<\Gamma(s)x\omega_s(x)\leq 2(2\pi)^{s-1}m!G\left(\frac{m+5}{2},1,x\right),
\]
where \(m=[s]\) and
\[
G(\alpha,\beta,x)=|\Gamma(\alpha+ix)\Gamma(\beta+ix)|^2.
\]
The layout of the paper is as follows:
\S1. Introduction (\(\frac{1}{2}\) pages)
\S2. Consequences of an integral representation of \(\zeta(s,a)\) (\(1\frac{1}{2}\) pages)
\S3. Bounds for the weight \(\omega_s(x)\) (\(3\) pages)
\S4. Wilson's polynomials (\(1\frac{1}{2}\) pages)
\S5. A bound for the Padé approximant of \(\Phi_s(z)\) (\(2\) pages)
\S6. Proofs of Theorem 1 and Corollary 1 (\(\frac{1}{2}\) page)
\S7. The case \(s\) real (negative)
In this section the main results given above are generalized to the case \(s<0\)
\S8. The case \(a=1\) and \(s\in\mathbb{N}\) (\(1\) page)
References (\(10\) items)
Reviewer: Marcel G. de Bruin (Heemstede)A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functionshttps://www.zbmath.org/1475.111532022-01-14T13:23:02.489162Z"Adam, Alexander"https://www.zbmath.org/authors/?q=ai:adam.alexander"Pohl, Anke"https://www.zbmath.org/authors/?q=ai:pohl.anke-dLet \(\Gamma\) be a cofinite or non-cofinite Hecke triangle group, and \(\chi\) be a finite dimensional unitary representation of \(\Gamma\). For \(\Re(s)>0\), we denote by \(\mathcal L_s^{\mathrm{slow},-}\) and \(\mathcal L_s^{\mathrm{fast,-}}\) (resp. \(\mathcal L_s^{\mathrm{slow},+}\) and \(\mathcal L_s^{\mathrm{fast,+}}\)) the odd (resp. even) parts of the associate families of slow and fast transfer operators, respectively.
The authors prove that the real analytic eigenfunctions with eigenvalue 1 of \(\mathcal L_s^{\mathrm{slow},+}\) (resp. \(\mathcal L_s^{\mathrm{slow},-}\)) that satisfy a certain growth condition are isomorphic to the eigenfunctions with eigenvalue 1 of \(\mathcal L_s^{\mathrm{fast},+}\) (resp. \(\mathcal L_s^{\mathrm{fast},-}\)). It is remarkable that the proof does not depend on the Selberg theory.
Reviewer: Shin-ya Koyama (Yokohama)Joint discrete universality for \(L\)-functions from the Selberg class and periodic Hurwitz zeta-functionshttps://www.zbmath.org/1475.111542022-01-14T13:23:02.489162Z"Balčiūnas, Aidas"https://www.zbmath.org/authors/?q=ai:balciunas.aidas"Macaitienė, Renata"https://www.zbmath.org/authors/?q=ai:macaitiene.renata"Šiaučiūnas, Darius"https://www.zbmath.org/authors/?q=ai:siauciunas.dariusIt is known from the pioneering work of Voronin that the shifts of some zeta and \(L\)-functions approximate a wide class of analytic functions. This property is termed as universality, which also extends to simultaneous approximation of a collection of analytic functions by a collection of zeta functions. The paper under review establishes joint universality theorems of this nature, which realize simultaneous approximation of a collection of analytic functions from a wide class by the shifts of two types of zeta functions. The first type is a subclass of Dirichlet series, introduced by Steuding, of the Selberg class \(\mathcal{S}\). The second type is the periodic Hurwitz zeta functions. The precise statements of the theorems are technical and can be found in the Introduction of the paper. The method of the paper is of probabilistic nature, and uses weak convergence of certain probability measures related to the above zeta functions.
Reviewer: Dongwen Liu (Zhejiang)On primeness of the Selberg zeta-functionhttps://www.zbmath.org/1475.111552022-01-14T13:23:02.489162Z"Garunkštis, Ramūnas"https://www.zbmath.org/authors/?q=ai:garunkstis.ramunas"Steuding, Jörn"https://www.zbmath.org/authors/?q=ai:steuding.jornThe paper under review studies the Selberg zeta-function \(Z(s)\) associated with a compact Riemann surface of genus \(g\). The main results states that \(Z(s)\) is pseudo-prime and right-prime. More precisely, for every decomposition \(Z(s)=f(h(s))\) with \(f\) meromorphic and \(h\) entire (or \(h\) meromorhic when \(f\) is rational), the following hold:
(1) \(f\) is rational or \(h\) is a polynomial;
(2) \(h\) is linear whenever \(f\) is transcendental (noting a typo in the Definition, p. 452).
Moreover, if \(f\) is rational and \(h\) is meromorphic, then \(f\) is a polynomial of degree \(k\) where \(k\) divides \(2g-2\), and \(h\) is entire.
Reviewer: Dongwen Liu (Zhejiang)Stieltjes constants of \(L\)-functions in the extended Selberg classhttps://www.zbmath.org/1475.111562022-01-14T13:23:02.489162Z"Inoue, Shōta"https://www.zbmath.org/authors/?q=ai:inoue.shota"Eddin, Sumaia Saad"https://www.zbmath.org/authors/?q=ai:saad-eddin.sumaia"Suriajaya, Ade Irma"https://www.zbmath.org/authors/?q=ai:suriajaya.ade-irmaSummary: Let \(f\) be an arithmetic function and let \(\mathcal{S}^\#\) denote the extended Selberg class. We denote by \(\mathcal{L}(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}\) the Dirichlet series attached to \(f\). The Laurent-Stieltjes constants of \({\mathcal{L}}(s)\), which belongs to \(\mathcal{S}^\#\), are the coefficients of the Laurent expansion of \(\mathcal{L}\) at its pole \(s=1\). In this paper, we give an upper bound of these constants, which is a generalization of many known results.Upper bounds of some special zeros for the Rankin-Selberg \(L\)-functionhttps://www.zbmath.org/1475.111572022-01-14T13:23:02.489162Z"Bllaca, Kajtaz H."https://www.zbmath.org/authors/?q=ai:bllaca.kajtaz-hSummary: In this paper, we prove some conditional results about the order of zero at central point \(s=1/2\) of the Rankin-Selberg \(L\)-function \(L(s,\pi_f\times\tilde{\pi}'_f)\). Then, we give an upper bound for the height of the first zero with positive imaginary part of \(L(s,\pi_f\times\tilde{\pi}'_f)\). We apply our results to automorphic \(L\)-functions.Uniqueness results for a class of \(L\)-functionshttps://www.zbmath.org/1475.111582022-01-14T13:23:02.489162Z"Dixit, Anup B."https://www.zbmath.org/authors/?q=ai:dixit.anup-bSummary: \textit{V. Kumar Murty} [The conference on \(L\) functions. Singapore: World Scientific. 165--174 (2007)] introduced a class of \(L\)-functions, namely the Lindelöf class, which contains the Selberg class and has a ring structure attached to it. In this paper, we establish some results on the a-value distribution of elements on a subclass of the Lindelöf class. As a corollary, we also prove a uniqueness theorem in the Selberg class.
For the entire collection see [Zbl 1403.11002].A Bohr-Jessen type theorem for the Epstein zeta-function. II.https://www.zbmath.org/1475.111592022-01-14T13:23:02.489162Z"Laurinčikas, Antanas"https://www.zbmath.org/authors/?q=ai:laurincikas.antanas"Macaitienė, Renata"https://www.zbmath.org/authors/?q=ai:macaitiene.renataIn the paper, value-distribution of the Epstein zeta-function \(\zeta(s;Q)\), \(s=\sigma+it\), on the complex plane \(\mathbb C\) is studied.
For positive definite quadratic \(n \times n\) matrix \(Q\), let \(Q[{\underline{x}}]={\underline x}^TQ{\underline x}\), \({\underline x}\in {\mathbb Z}^n\). Then the Epstein zeta-function \(\zeta(s;Q)\) is defined by the series \[ \zeta(s;Q)=\sum_{{\underline x}\in {\mathbb Z}^n\setminus\{{\underline 0}\}}(Q[{\underline x}])^{-s}, \quad \sigma>\frac{n}{2}, \] and it has analytic continuation to whole \(s\)-plane, except for a simple pole at \(s=\frac{n}{2}\) with residue \(\frac{\pi^{n/2}}{\Gamma(n/2)\sqrt{{\mathrm{det}}Q}}\). Then the discrete limit theorem for the function \(\zeta(s;Q)\) is proved, i.e., it is shown that, for fixed \(\sigma>\frac{n-1}{2}\), \[ \frac{1}{N+1}\# \big\{0 \leq k \leq N: \zeta(\sigma+ikh;Q)\in A\big\}, \quad A \in {\mathcal B}({\mathbb{C}}), \] converges weakly to explicitly given probabilty measure as \(N \to \infty\). Note that two types of fixed common difference \(h> 0\) of arithmetic progression \(\{kh: k\in {\mathbb{N}}\}\) are studied: (i) when \(h\) is such that the number \(\exp\{\frac{2 \pi m}{h}\}\) is irrational for all \(m \in {\mathbb{N}}\), and (ii) when \(h\) is not of (i) type.
Reviewer: Roma Kačinskaitė (Kaunas)Distribution and non-vanishing of special values of \(L\)-series attached to Erdős functionshttps://www.zbmath.org/1475.111602022-01-14T13:23:02.489162Z"Pathak, Siddhi S."https://www.zbmath.org/authors/?q=ai:pathak.siddhi-sLet \(q\) be a positive integer and \(f\) be a \(q\)-periodic arithmetic function such that \(f(n)\in\{-1,1\}\) when \(q\not|n\) and \(f(n)=0\) otherwise. Erdős conjectured that \(\sum_{n=1}^{\infty}f(n)/n\neq 0\) whenever this series converges (cf. [\textit{A. E. Livingston}, Can. Math. Bull. 8, 413--432 (1965; Zbl 0129.02801)]). The author shows here that Erdős conjecture holds with ``probability'' 1. This improves on a result by \textit{T. Chatterjee} and \textit{M. R. Murty} [Pac. J. Math. 275, No. 1, 103--113 (2015; Zbl 1333.11084)].
A rational-valued \(q\)-period arithmetic function \(f\) is called an Erdős function \(\mod q\) if \(f(n)\in\{-1,1\}\) when \(q\not|n\) and \(f(n)=0\) otherwise and \(\sum_{a=1}^{q}f(a)=0\). The \(L\)-series associated to \(f\) is \(L(s,f)=\sum_{n=1}^{\infty}f(n)/n^s\). The author also obtains the characteristic function of the limiting distribution of \(L(k,f)\) for any positive integer \(k\) and Erdős function \(f\) with the same parity as \(k\).
Reviewer: Jasson Vindas (Gent)Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbershttps://www.zbmath.org/1475.111612022-01-14T13:23:02.489162Z"Stewart, Seán Mark"https://www.zbmath.org/authors/?q=ai:stewart.sean-markSummary: In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers \(H_{2n}\) are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers \(H_n \). As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung
\[
\sum\limits_{n = 1}^\infty \biggl(\frac{H_n}{n}\biggr)^2=\frac{17 \pi^4}{360},
\]
together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approachhttps://www.zbmath.org/1475.111622022-01-14T13:23:02.489162Z"Broucke, Frederik"https://www.zbmath.org/authors/?q=ai:broucke.frederik"Debruyne, Gregory"https://www.zbmath.org/authors/?q=ai:debruyne.gregory"Vindas, Jasson"https://www.zbmath.org/authors/?q=ai:vindas.jassonThis is an extension of the paper [the last two authors, Proc. Am. Math. Soc. 146, No. 12, 5097--5103 (2018; Zbl 1454.11170)]. In the latter, it is shown that there is no better estimate for the remainder in the Wiener-Ikehara Theorem, even if one assumes analytic continuation to a half-plane. The proof is non-constructive and based on principles of functional analysis.
The present paper answers the natural question for an explicit construction of a counterexample. Using a similar construction, an analogous counterexample to a sharpening of the Ingham-Karamata Theorem is given.
Reviewer: Anton Deitmar (Tübingen)A new circle method attack on twin primeshttps://www.zbmath.org/1475.111642022-01-14T13:23:02.489162Z"Mozzochi, C. J."https://www.zbmath.org/authors/?q=ai:mozzochi.charles-jSummary: We present a new plan of attack for estimating the minor arcs. Additionally, in the process, we do not assume GRH.Uniform analytic properties of representation zeta functions of finitely generated nilpotent groupshttps://www.zbmath.org/1475.200592022-01-14T13:23:02.489162Z"Dung, Duong H."https://www.zbmath.org/authors/?q=ai:dung.duong-hoang"Voll, Christopher"https://www.zbmath.org/authors/?q=ai:voll.christopherSummary: Let \( G\) be a finitely generated nilpotent group. The representation zeta function \( \zeta _G(s)\) of \( G\) enumerates twist isoclasses of finite-dimensional irreducible complex representations of \( G\). We prove that \( \zeta _G(s)\) has rational abscissa of convergence \( \alpha (G)\) and may be meromorphically continued to the left of \( \alpha (G)\) and that, on the line \( \{s\in \mathbb{C} \mid \mathrm {Re}(s) = \alpha (G)\}\), the continued function is holomorphic except for a pole at \( s=\alpha (G)\). A Tauberian theorem yields a precise asymptotic result on the representation growth of \( G\) in terms of the position and order of this pole.
We obtain these results as a consequence of a result establishing uniform analytic properties of representation zeta functions of torsion-free finitely generated nilpotent groups of the form \( \mathbf {G}(\mathcal {O})\), where \( \mathbf {G}\) is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring \( \mathcal {O}\) of integers of a
number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of \( \mathbf {G}\), independent of \( \mathcal {O}\).An inequality for the modified Selberg zeta-functionhttps://www.zbmath.org/1475.300012022-01-14T13:23:02.489162Z"Belovas, Igoris"https://www.zbmath.org/authors/?q=ai:belovas.igorisSummary: We consider the absolute values of the modified Selberg zeta-function at places symmetric with respect to the critical line. We prove an inequality for the modified Selberg zeta-function in a different way, reproving and extending the result of Garunkštis and Grigutis and completing the extension of a result of Belovas and Sakalauskas.On the approximation of analytic functions by shifts of an absolutely convergent Dirichlet serieshttps://www.zbmath.org/1475.300892022-01-14T13:23:02.489162Z"Jasas, M."https://www.zbmath.org/authors/?q=ai:jasas.m"Laurinčikas, A."https://www.zbmath.org/authors/?q=ai:laurincikas.antanas"Šiaučiūnas, D."https://www.zbmath.org/authors/?q=ai:siauciunas.dariusSummary: A theorem dealing with the approximation of analytic functions in the strip \(\{s\in \mathbb{C}: 1/2< \operatorname{Re} s<1\}\) by shifts of an absolutely convergent Dirichlet series close to a periodic zeta-function with multiplicative coefficients is proved.