Recent zbMATH articles in MSC 11Mhttps://zbmath.org/atom/cc/11M2023-12-07T16:00:11.105023ZWerkzeugBook review of: G. Helmberg, Analytische Zahlentheorie. Rund um den Primzahlsatzhttps://zbmath.org/1522.000182023-12-07T16:00:11.105023Z"Baxa, C."https://zbmath.org/authors/?q=ai:baxa.christophReview of [Zbl 1404.11001].Book review of: P. J. Nahin, In pursuit of zeta-3. The world's most mysterious unsolved math problemhttps://zbmath.org/1522.001112023-12-07T16:00:11.105023Z"Simoson, Andrew"https://zbmath.org/authors/?q=ai:simoson.andrew-jReview of [Zbl 1485.11002].Book review of: J.-M. de Koninck and N. Doyon, The life of primes in 37 episodeshttps://zbmath.org/1522.001302023-12-07T16:00:11.105023Z"Watkins, John J."https://zbmath.org/authors/?q=ai:watkins.john-jReview of [Zbl 1480.11001].Book review of: P. Garrett, Modern analysis of automorphic forms by example. Volume 1 and 2https://zbmath.org/1522.001372023-12-07T16:00:11.105023Z"Young, Matthew P."https://zbmath.org/authors/?q=ai:young.matthew-pReview of [Zbl 1459.11002; Zbl 1487.11002].On the splitting conjecture in the hybrid model for the Riemann zeta functionhttps://zbmath.org/1522.110852023-12-07T16:00:11.105023Z"Heap, Winston"https://zbmath.org/authors/?q=ai:heap.winston-pFrom the random matrix theory \textit{J. P. Keating} and \textit{N. C. Snaith} [Commun. Math. Phys. 214, 57--89 (2000; Zbl 1051.11048)] conjectured the asymptotic formulas for the moments of the Riemann zeta function on the critical line. Here the constants appearing in the asymptotic formulas consist of an arithmetic part, which is expressed in terms of a product over the prime numbers, and a part from the random matrix theory. To explain the arithmetic factor, \textit{S. M. Gonek}, \textit{C. P. Hughes} and \textit{J. P. Keating} [Duke Math. J. 136, 507-549 (2007; Zbl 1171.11049)] expressed \(\zeta(s)\) as the hybrid Euler-Hadamard product of the form
\[
\zeta(\tfrac{1}{2}+it)=P_X(\tfrac{1}{2}+it)Z_X(\tfrac{1}{2}+it)(1+(\log X)^{-1})
\]
for \(2\leq X\leq t^{1/3}\) and large \(t\). Here \(P_X(s)\) and \(Z_X(s)\) are given by
\[
P_X(s)=\exp\left(\sum_{2\leq n\leq X}\frac{\Lambda(n)}{n^s\log n}\right), \quad Z_X(s)=\exp\left(-\sum_{\rho}U((s-\rho)\log X)\right),
\]
where \(\Lambda(n)\) is the von Mangoldt function, \(\rho\) runs over the nontrivial zeros and \(U\) is given by a certain integral. The splitting conjecture by Gonek-Hughes-Keating is stated as
\[
\frac{1}{T}\int_T^{2T}\left|P_X(\tfrac{1}{2}+it)Z_X(\tfrac{1}{2}+it)\right|^{2k}dt
\sim\left(\frac{1}{T}\int_T^{2T}\left|P_X(\tfrac{1}{2}+it)\right|^{2k}dt\right) \left(\frac{1}{T}\int_T^{2T}\left|Z_X(\tfrac{1}{2}+it)\right|^{2k}dt\right)
\]
as \(T\to\infty\) with \(X\ll(\log T)^{2-\varepsilon}\) for each \(\varepsilon>0\) and \(k>-1/2\).
In this paper the author shows that the splitting conjecture holds to order for a larger range of \(X\) under the Riemann hypothesis. Namely, under the Riemann hypothesis
\[
\frac{1}{T}\int_T^{2T}\left|P_X(\tfrac{1}{2}+it)Z_X(\tfrac{1}{2}+it)\right|^{2k}dt
\asymp\left(\frac{1}{T}\int_T^{2T}\left|P_X(\tfrac{1}{2}+it)\right|^{2k}dt\right) \left(\frac{1}{T}\int_T^{2T}\left|Z_X(\tfrac{1}{2}+it)\right|^{2k}dt\right)
\]
holds as \(X\leq (\log T)^{\theta_k-\varepsilon}\), where \(\theta_k=2\sqrt{1+1/2|k|}\). This paper also improves the splitting conjecture for \(k=1,2\), which was confirmed by Gonek et al. Namely, the author shows it in larger ranges of \(X\), unconditionally and under the Riemann hypothesis.
Reviewer: Hirotaka Akatsuka (Otaru)A new reason for doubting the Riemann hypothesishttps://zbmath.org/1522.110862023-12-07T16:00:11.105023Z"Blanc, Philippe"https://zbmath.org/authors/?q=ai:blanc.philippe|blanc.philippe.2|blanc.philippe.1Summary: Assuming computations of the Riemann zeta function exhibit its true behavior, we get, under the Riemann hypothesis, a bound for a linear combination of odd order derivatives of Hardy's \(Z\)-function evaluated at \(T + a\) and \(T - a\) where \(T \pm a\) are some well chosen inflection points of \(Z\). This bound, which only holds for \(T \pm a\) beyond the computational capabilities of modern computers, suggests that Riemann hypothesis is not true. The key element in our argument is an identity which links the zeroes of a function \(f\) defined on the interval \([-a,a]\) and the values of its derivatives of odd order at \(\pm a\).Erdős-macintyre type theorem's for multiple Dirichlet series: exceptional sets and open problemshttps://zbmath.org/1522.110872023-12-07T16:00:11.105023Z"Bandura, A. I."https://zbmath.org/authors/?q=ai:bandura.a-i"Salo, T. M."https://zbmath.org/authors/?q=ai:salo.tetyana-mykhailivna"Skaskiv, O. B."https://zbmath.org/authors/?q=ai:skaskiv.oleg-bThe Wiman-Valiron inequality is a classical inequality of non-constant entire functions of the form
\[
f(z)=\sum_{n=0}^{\infty}a_nz^n.
\]
For every \(\varepsilon>0\) there exists an exceptional set \(E\subset(1,\infty)\) such that for \(r\in(1,\infty)\setminus E\) the estimate
\[
\max_{|z|=r} |f(z)|\le \mu_f(r)(\log\mu_f(r))^{\frac{1}{2}+\varepsilon}
\]
holds, where \(\mu_f(r)=\max_{n\ge0}|a_n|r^n\). It is well known that the exceptional set \(E\) in the classical Wiman-Valiron inequality is a union of closed intervals \(E=\cup[a_n, b_n]\) of finite logarithmic measures, that is, \(\int_Ed\ \log r <\infty\). Let \(E=\cup[a_n, b_n]\) be an arbitrary set of finite logarithmic measures. Ostrovskii raised the question whether there exists an entire function such that the set \(E\) is an exceptional set in the classical Wiman-Valiron inequality for given \(\varepsilon\).
In this paper, the authors consider this problem for entire functions on \(\mathbb C^p\) represented by absolutely convergent multiple Dirichlet series in the entire space \(\mathbb C^p\).
Reviewer: Wataru Takeda (Tōkyō)A generalization of quasi-shuffle algebras and an application to multiple zeta valueshttps://zbmath.org/1522.110882023-12-07T16:00:11.105023Z"Keilthy, Adam"https://zbmath.org/authors/?q=ai:keilthy.adamThe quasi-shuffle structure of multiple zeta values was studied in detail by \textit{M. E. Hoffman} [J. Algebr. Comb. 11, No. 1, 49--68 (2000; Zbl 0959.16021)]. Recently, \textit{M. Hirose} and \textit{N. Sato} [``Block shuffle identities for multiple zeta values'', Preprint, \url{arXiv:2206.03458}]. defined the block shuffle product and established a new family of relations among multiple zeta values. In this paper, the author defines a commutative algebra structure on the space of non-commutative polynomials, which generalized the block shuffle product of Hirose and Sato. Moreover, the author shows that this quasi-shuffle algebra is isomorphic to the standard shuffle algebra over rational field.
Reviewer: Jiangtao Li (Beijing)Square integrals of the logarithmic derivatives of Selberg's zeta functions in the critical striphttps://zbmath.org/1522.110892023-12-07T16:00:11.105023Z"Hashimoto, Yasufumi"https://zbmath.org/authors/?q=ai:hashimoto.yasufumiLet \(\Gamma\) be a (not necessarily congruence) subgroup of the modular group or a cocompact arithmetic subgroup of \(\mathrm{SL}_2(\mathbb R)\) coming from a quaternion algebra. The main theorem of this article improves the mean value estimate of the square of the logarithmic derivative of the Selberg zeta function \(Z_\Gamma(s)\) in the critical strip, which the author obtained in his previous work.
Indeed he proves this time that for \(1/2<\sigma<1\) it holds that
\[
\frac1T\int_1^T\left|\frac{Z_\Gamma'(\sigma+it)}{Z_\Gamma(\sigma+it)}\right|^2dt \ll_{\varepsilon,\Gamma} T^{\max(0,5-6\sigma+\varepsilon)}\qquad(T\to\infty).
\]
Reviewer: Shin-ya Koyama (Yokohama)Inverse problem for a monoid with an exponential sequence of primeshttps://zbmath.org/1522.110902023-12-07T16:00:11.105023Z"Dobrovol'skiĭ, Nikolaĭ Nikolaevich"https://zbmath.org/authors/?q=ai:dobrovolskii.n-n"Rebrova, Irina Yur'evna"https://zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Dobrovol'skiĭ, Nikolaĭ Mikhaĭlovich"https://zbmath.org/authors/?q=ai:dobrovolskii.n-mSummary: In this paper, for an arbitrary monoid \({M(PE)}\) with an exponential sequence of primes \(PE\) of type \(q\), the inverse problem is solved, that is, finding the asymptotic for the distribution function of elements of the monoid \({M(PE)}\), based on the asymptotic distribution of primes of the sequence of primes \(PE\) of type \(q\).
To solve this problem, we introduce the concept of an arbitrary exponential sequence of natural numbers of the type \(q\) and consider the monoid generated by this sequence. Using two homomorphisms of such monoids, the density distribution problem is reduced to the additive Ingham problem.
It is shown that the concept of power density does not work for this class of monoids. A new concept of \(C\) logarithmic \(\theta \)-power density is introduced.
It is shown that any monoid \({M(PE)}\) for an arbitrary exponential sequence of primes \(PE\) of type \(q\) has \(C\) logarithmic \(\theta \)-power density with \(C=\pi\sqrt{\frac{2}{3\ln q}}\) and \(\theta=\frac{1}{2} \).A multidimensional analog of the Weierstrass \(\zeta \)-function in the problem of the number of integer points in a domainhttps://zbmath.org/1522.110912023-12-07T16:00:11.105023Z"Tereshonok, Elena N."https://zbmath.org/authors/?q=ai:tereshonok.elena-n"Shchuplev, Alexey V."https://zbmath.org/authors/?q=ai:shchuplev.alexey-vSummary: A multidimensional analog of the Weierstrass \(\zeta \)-function in \(\mathbb{C}^n\) is a differential \((0,n-1)\)-form with singularities in the points of the integer lattice \(\Gamma\subset\mathbb{C}^n\). Using this form we construct a \(\Gamma \)-invariant \((n,n-1)\)-form \(\tau(z)\wedge dz\). The integral of this form over a domain's boundary is equal to difference between the number of integer points in the domain and its volume.The weak Gram law for Hecke \(L\)-functionshttps://zbmath.org/1522.110922023-12-07T16:00:11.105023Z"Weishäupl, Sebastian"https://zbmath.org/authors/?q=ai:weishaupl.sebastianThe Hardy \(Z\)-function is \(Z_\zeta(t) := e^{i \vartheta_\zeta(t)} \zeta(\tfrac{1}{2} + it)\), where \(\vartheta_\zeta(t)\) is the Riemann-Siegel theta function. The \(e^{i \vartheta_\zeta(t)}\) factor has the effect of making \(Z(t)\) real-valued when \(t\) is real, and critical zeros of \(\zeta(\tfrac{1}{2} + it)\) agree with zeros of \(Z_\zeta(t)\).
The Gram law states roughly that zeros of \(Z_\zeta\) tend to alternate with \emph{Gram points}, which are points \(t_n\) where \(\vartheta_\zeta(t_n) = n \pi\). More precise versions are in Titchmarsh's tome on \emph{the theory of the Riemann zeta-function}. As \(N \to \infty\) and for any \(M \geq 0\), Titchmarsh shows
\begin{align*}
\sum_{n = M}^N Z_\zeta(t_{2n}) &= 2N + O(N^{\frac{3}{4}} \log^{\frac{3}{4}} N), \\
\sum_{n = M}^N Z_\zeta(t_{2n + 1}) &= -2N + O(N^{\frac{3}{4}} \log^{\frac{3}{4}} N).
\end{align*}
As \(Z_\zeta(t)\) is real valued, this implies that there are infinitely many intervals \((t_n, t_{n+1}]\) containing an odd number of zeros of \(\zeta(\tfrac{1}{2} + it)\). This is also sometimes called the weak Gram law.
This paper proves the analogous result for \(L\)-functions associated to level \(1\) cuspidal Hecke eigenforms of weight \(k \geq 12\). One can construct a real-valued function \(Z_L(t)\) associated to \(L(s, f)\) just a for \(Z_\zeta(t)\). The analogous result is Corollary 2, which states that for \(M\) sufficiently large and as \(N \to \infty\),
\begin{align*}
\sum_{n = M}^N Z_L(t_{2n}) &= 2N + O_{L, \varepsilon}(N^{\frac{3}{4} + \varepsilon}), \\
\sum_{n = M}^N Z_L(t_{2n + 1}) &= -2N + O_{L, \varepsilon}(N^{\frac{3}{4} + \varepsilon}).
\end{align*}
This follows from a weighted version, which is presented as Theorem 1 and which states that as \(T \to \infty\),
\begin{align*}
\sum_{T < t_{2n} \leq 2T} \omega(t_{2n}) Z_L(t_{2n}) &= \frac{1}{\pi} T + O_{L, \varepsilon}(T^{\frac{3}{4} + \varepsilon}), \\
\sum_{T < t_{2n+1} \leq 2T} \omega(t_{2n + 1}) Z_L(t_{2n+1}) &= -\frac{1}{\pi} T + O_{L, \varepsilon}(T^{\frac{3}{4} + \varepsilon}),
\end{align*}
where the weight function is given by \(\omega(t) = \log(t/2 \pi)^{-1}\).
To prove these results, the author works with contour integration and integral transforms directly; this is in contrast to Titchmarsh, who can apply the particularly well-behaved approximate functional equations available for the zeta function.
The results in this paper can be generalized to cusp forms on congruence subgroups with mostly notational differences.
Reviewer: David Lowry-Duda (Providence)Integrality of \(v \)-adic multiple zeta valueshttps://zbmath.org/1522.111182023-12-07T16:00:11.105023Z"Chen, Yen-Tsung"https://zbmath.org/authors/?q=ai:chen.yen-tsungReal multiple zeta values MZVs are the real numbers
\[
\zeta(s_1,\ldots,s_r):=\sum_{n_1>\cdots>n_r\geq 1}\frac 1{ n_1^{s_1}\cdots n_r^{s_r}}\in {\mathbb R}^*,
\]
where \(s_1\geq 2\). Set \({\mathfrak s}:=(s_1,\ldots,s_r)\in {\mathbb N}^r\). The \textit{depth} of \({\mathfrak s}\) is defined as \(\mathrm{dep}(\mathfrak s)=r\), \({\mathrm{wt}}({\mathfrak s})= \sum_{i=1}^r s_i\) is called the \textit{weight} and \({\mathrm{ht}}( {\mathfrak s})=|\{i\mid s_i\neq 1\}|\) is called the \textit{height}. Let \({\mathrm{Li}}_{(s_1,\ldots,s_r)}(z)=\sum_{n_1>n_2>\cdots >n_r\geq 1} \frac{z^{n_1}}{n_1^{s_1} \cdots n_r^{s_r}}\). Then \(\zeta(s_1,\ldots,s_r)={\mathrm{Li}}_{(s_1,\ldots,s_r)}(z)|_{z=1}\). Let \({\mathrm{Li}}_{(s_1,\ldots,s_r)}(z)_p\) be the \(p\)-adic function defined by the same series on \({\mathbb C}_p\). The \(p\)-adic MZV \(\zeta(s_1,\ldots,s_r)_p\) is defined by taking certain limit \(z\to 1\).
Akagi-Hirose-Yasuda and
\textit{A. Chatzistamatiou} [J. Algebra 474, 240--270 (2017; Zbl 1422.11234)] showed that every \(p\)-adic MZV is a \(p\)-adic integer. Furthermore, for all but finitely many primes \(p\), the \(p\)-adic valuation of \(p\)-adic MZVs is greater than \({\mathrm{wt}}({\mathfrak s})\). They also proved that for each integer \(w\in{\mathbb N}\), if \(\frac{1-X^2}{1-X^2-X^3}=\sum_{ w\geq 0} (d_{w,{\mathcal A}}) X^w\), then \(\dim_{\mathbb Q} Z_{ w,{\mathcal A}}\leq d_{w,{\mathcal A}}\), where \({\mathcal A}:= \big(\prod_p{\mathbb Z}/p{\mathbb Z}\big)\otimes_{\mathbb Z} {\mathbb Q}\), where \(p\) runs over all primes \(p\) and \(Z_{ w,{\mathcal A}}\) is the \({\mathbb Q}\)-vector space generated by all \textit{finite multiple zeta values} (FMZVs) \(\zeta_{\mathcal A}(s_1,\ldots,s_r):=(\zeta_{\mathcal A}(s_1,\ldots,s_r)_p)_p\in {\mathcal A}\) of weight \(w\).
The main purpose of the article under review is to study the function field analogue of the results of Akagi-Hirose-Yasuda. Let \(A:={\mathbb F}_q[\theta]\), \(k={\mathbb F}_q(\theta)\), and \(k_{\infty}\) the completion of \(k\) at \(\infty\). The \(\infty\)-adic MZV is defined as \(\zeta_A(s_1,\ldots,s_r):=\sum\frac 1{a_1^{s_1}\cdots a_r^{s_r}}\in k_{\infty}\), with \((a_1,\ldots, a_r)\in A^r\), \(a_i\) monic and \(\deg_{\theta} a_1>\deg_{\theta} a_2>\cdots>\deg_{\theta} a_r\). Let
\[
{\mathrm{Li}}^*_{(s_1,\ldots,s_r)}(z_1,\ldots,z_r):= \sum_{i_1\geq i_2 \geq \cdots \geq i_r\geq 0}\frac {z_1^{q^{i_1}}\cdots z_r^{q^{i_r}}} {L_{i_1}^{s_1}\cdots L_{i_r}^{s_r}},
\]
where \(L_0:=1\) and \(L_i:=(\theta-\theta^q)\cdots (\theta - \theta^{q^i})\), \(i\geq 1\). Let \(v\) be a fixed finite place of \(k\) and consider \({\mathrm{Li}}^*_{(s_1,\ldots,s_r)}(z_1,\ldots,z_r)_v\) as a \(v\)-adic function defined by the same series on \({\mathbb C}_v^r\), where \({\mathbb C}_v\) is the completion of an algebraic closure of the completion of \(k\) at \(v\). The corresponding \(v\)-adic MZVs are the main objects of study in this work.
The main result, Theorem 4.2.1, states that if \({\mathfrak s}= (s_1,\ldots,s_r)\in{\mathbb N}^r\), \(q_v\) is the cardinality of the residue field \(A/vA\), and \(B_{w,v}:=\min_{n\geq 0}\{q_v^n -n\cdot w\}\), then \({\mathrm{ord}}_v(\zeta_A({\mathfrak s})_v)\geq B_{{\mathrm{wt}}({\mathfrak s}), v}-\frac{{\mathrm{wt}}({\mathfrak s}) -{\mathrm{dep}}({\mathfrak s})-{\mathrm{ht}}({\mathfrak s})}{q_v-1}\). In particular \(\zeta_A({\mathfrak s})_v\in A_v\) if \(q_v\geq {\mathrm{wt}}({\mathfrak s})\).
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)Partial zeta functions, partial exponential sums, and \(p\)-adic estimateshttps://zbmath.org/1522.111272023-12-07T16:00:11.105023Z"Bertram, Noah"https://zbmath.org/authors/?q=ai:bertram.noah"Deng, Xiantao"https://zbmath.org/authors/?q=ai:deng.xiantao"Haessig, C. Douglas"https://zbmath.org/authors/?q=ai:haessig.c-douglas"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.15|li.yan.7|li.yan.21|li.yan.12|li.yan.16|li.yan.25|li.yan.28|li.yan.62|li.yan.11|li.yan.5|li.yan.10|li.yan|li.yan.19|li.yan.14|li.yan.9|li.yan.24|li.yan.2Summary: Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, we give Chevalley-Warning type bounds for partial zeta functions and \(L\)-functions associated to partial exponential sums.Machine learning invariants of arithmetic curveshttps://zbmath.org/1522.140042023-12-07T16:00:11.105023Z"He, Yang-Hui"https://zbmath.org/authors/?q=ai:he.yang-hui"Lee, Kyu-Hwan"https://zbmath.org/authors/?q=ai:lee.kyu-hwan"Oliver, Thomas"https://zbmath.org/authors/?q=ai:oliver.thomasIn the present paper, the authors show that an machine learning classifier can be trained to predict the rank and the torsion order of an elliptic curve or a genus two curve with high precision when the curve is represented by a few hundred coefficients of its \(L\)-function. In particular, for elliptic curves, the torsion order and the number of integral points are determined almost perfectly by machine learning classifiers. Among the discrete invariants appearing in the BSD conjecture, only the order of the Tate-Shafarevich group seems to be out of reach with their approach of using a finite number of \(L\)-function coefficients.
Reviewer: Lei Yang (Beijing)Joint distribution of the cokernels of random \(p\)-adic matriceshttps://zbmath.org/1522.150302023-12-07T16:00:11.105023Z"Lee, Jungin"https://zbmath.org/authors/?q=ai:lee.junginLet \(P_1(t),\dots,P_\ell(t)\in\mathbb Z_p[t]\) be monic polynomials such that their reductions modulo \(p\) in \(\mathbb F_p[t]\) are distinct and irreducible. Let \(H_j\) be a finite module over \(R_j:=\mathbb Z_p[t]/(P_j(t))\) for each \(1\le j\le \ell\). In the first main theorem of this article, the author proves that
\[
\lim_{n\to\infty}\operatorname*{Prob}_{A\in M_n(\mathbb Z_p)} (\mathrm{cok}(P_j(A))\cong H_j \text{ for }1\le j\le \ell)
=\prod_{j=1}^\ell\left(\frac1{|\mathrm{Aut}_{R_j}(H_j)|} \prod_{i=1}^\infty(1-p^{-i\mathrm{deg}(P_j)})\right).
\]
This is a generalization of the theorem of \textit{G. Cheong} and \textit{N. Kaplan} [J. Algebra 604, 636--663 (2022; Zbl 1490.15051)] who proved the case of \(\mathrm{deg}(P_j)\le 2\) for each \(j\).
Let \(\{B_n\}\) be any sequence of matrices such that \(B_n\in M_n(\mathbb Z_p)\) and that
\[
\lim_{n\to\infty}(n-\log_p n-r_p(\mathrm{cok}(B_n)))=\infty
\]
with \(r_p(M):=\mathrm{rank}_{\mathbb F_p}(M/pM)\). In the second main theorem the author proves that for any finite abelian \(p\)-groups \(H_1\) and \(H_2\), there holds
\[
\lim_{n\to\infty}\operatorname*{Prob}_{A\in M_n(\mathbb Z_p)}(\mathrm{cok}(A)\cong H_1 \text{ and }\mathrm{cok}(A+B_n)\cong H_2)
=\prod_{j=1}^2\left(\frac1{|\mathrm{Aut_{\mathbb Z_p}(H_j)}|} \prod_{i=1}^\infty(1-p^{-i})\right).
\]
Reviewer: Shin-ya Koyama (Yokohama)A class of identities associated with Dirichlet series satisfying Hecke's functional equationhttps://zbmath.org/1522.330052023-12-07T16:00:11.105023Z"Berndt, Bruce C."https://zbmath.org/authors/?q=ai:berndt.bruce-c"Dixit, Atul"https://zbmath.org/authors/?q=ai:dixit.atul"Gupta, Rajat"https://zbmath.org/authors/?q=ai:gupta.rajat"Zaharescu, Alexandru"https://zbmath.org/authors/?q=ai:zaharescu.alexandruSummary: We consider two sequences \(a(n)\) and \(b(n)\), \(1\leq n<\infty\), generated by Dirichlet series of the forms
\[
\sum\limits_{n=1}^{\infty}\frac{a(n)}{\lambda_n^s}\quad\text{and}\quad \sum\limits_{n=1}^{\infty}\frac{b(n)}{\mu_n^s},
\]
satisfying a familiar functional equation involving the gamma function \(\Gamma(s)\). A general identity is established. Appearing on one side is an infinite series involving \(a(n)\) and modified Bessel functions \(K_{\nu}\), wherein on the other side is an infinite series involving \(b(n)\) that is an analogue of the Hurwitz zeta function. Six special cases, including \(a(n)=\tau (n)\) and \(a(n)=r_k(n)\), are examined, where \(\tau (n)\) is Ramanujan's arithmetical function and \(r_k(n)\) denotes the number of representations of \(n\) as a sum of \(k\) squares. All but one of the examples appear to be new.Dynamical zeta functions and the distribution of orbitshttps://zbmath.org/1522.370382023-12-07T16:00:11.105023Z"Pollicott, Mark"https://zbmath.org/authors/?q=ai:pollicott.markSummary: In this survey we will consider various counting and equidistribution results associated to orbits of dynamical systems, particularly geodesic and Anosov flows. Key tools in this analysis are appropriate complex functions, such as the zeta functions of Selberg and Ruelle, and Poincaré series. To help place these definitions and results into a broader context, we first describe the more familiar Riemann zeta function in number theory, the Ihara zeta function for graphs and the Artin-Mazur zeta function for diffeomorphisms.
For the entire collection see [Zbl 1439.14001].