Selberg zeta functions for spaces of higher rank

The Ihara-Selberg zeta function for PGL(3) and Hecke operators

We prove an approximation to an Ihara-formula for the zeta function of an arithmetic quotient of the Bruhat-Tits building of a p-adic PGL(3).

Simple functional equations for generalized Selberg zeta functions with Tate motives

We prove that for a compact locally symmetric Riemannian space $M$ of rank 1 there exist infinitely many automorphic Tate motives $f$ such that the generalized Selberg zeta function $Z_{M(f)}(s)$ satisfies a simple functional equation in the sense that it has no gamma factors.

Turing's method for the Selberg zeta-function

In one of his final research papers, Alan Turing introduced a method to certify the completeness of a purported list of zeros of the Riemann zeta-function. In this paper we consider Turing's method in the analogous setting of Selberg zeta-functions, and we demonstrate that it can be carried out rigorously in the prototypical case of the modular surface.

Zeros of the extended Selberg class zeta-functions and of their derivatives

Levinson and Montgomery proved that the Riemann zeta-function $ζ(s)$ and its derivative have approximately the same number of non-real zeros left of the critical line. R. Spira showed that $ζ'(1/2+it)=0$ implies $ζ(1/2+it)=0$. Here we obtain that in small areas located to the left of the critical line and near it the functions $ζ(s)$ and $ζ'(s)$ have the same number of zeros. We prove our result for more general zeta-functions from the extended Selberg class $S$. We also consider zero trajectories of a certain family of zeta-functions from $S$.

The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function

The present paper is a report on joint work with Alessandro Languasco and Alberto Perelli on our recent investigations on the Selberg integral and its connections to Montgomery's pair-correlation function. We introduce a more general form of the Selberg integral and connect it to a new pair-correlation function, emphasising its relations to the distribution of prime numbers in short intervals.

Converse theorems: from the Riemann zeta function to the Selberg class

This is an expanded version of the author's lecture at the XX Congresso U.M.I., held in Siena in September 2015. After a brief review of L-functions, we turn to the classical converse theorems of H.Hamburger, E.Hecke and A.Weil, and to some later developments. Finally we present several results on converse theorems in the framework of the Selberg class of L-functions.

塞尔伯格ζ函数在σ=1处具有二阶矩

Second moments of general Dirichlet series and Selberg zeta functions at $σ= 1$

本文证明有限共形富克斯群的塞尔伯格ζ函数在σ=1处二阶矩存在，借助Broucke和Hilberdink的方法规避了Landau针对一般狄利克雷级数的分离条件。

临界带内 Selberg zeta 函数对数导数的平方积分

Square integrals of the logarithmic derivatives of Selberg's zeta functions in the critical strip

本文在已有对数导数上界工作的基础上，研究 Selberg zeta 函数对数导数的平方积分在模群、余紧算术群及其子群情形下的增长性质。

Artin formalism for Selberg zeta functions of co-finite Kleinian groups

Let $Γ\backslash\mathbb H^3$ be a finite-volume quotient of the upper-half space, where $Γ\subset {\rm SL}(2,\mathbb C)$ is a discrete subgroup. To a finite dimensional unitary representation $χ$ of $Γ$ one associates the Selberg zeta function $Z(s;Γ;χ)$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tildeΓ$ is a finite index group extension of $Γ$ in ${\rm SL}(2,\mathbb C)$, and $π={\rm Ind}_Γ^{\tildeΓ}χ$ is the induced representation, then $Z(s;Γ;χ)=Z(s;\tildeΓ;π)$. In the second part of the paper we prove by a direct method the analogous identity for…

The Selberg Trace Formula and Selberg Zeta-Function for Cofinite Kleinian Groups with Finite Dimensional Unitary Representations: Stony Brook University PhD Thesis

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification point to the zeta function. In fact, if D is the ring of Eisenstein integers, then the Selberg zeta-function of PSL(2,D) contains ramification points.

The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification point to the zeta function. In fact, if D is the ring of Eisenstein integers, then the Selberg zeta-function of PSL(2,D) contains ramification points and is the sixth-root of a meromorphic function.

On the Selberg integral of the $k-$divisor function and the $2k-$th moment of the Riemann zeta function

We deeply appreciate the papers of Ivić on the links between the $2k-$th moments of the Riemann zeta function and, say, $d_k$, the $k-$divisor function. More specifically, both the one bounding the $2k-$th moment with a simple average of correlations of the $d_k$ (Palanga 1996 Conference Proceedings) and the more recent (arXiv:0708.1601v2 to appear on JTNB), which bounds the Selberg integral of $d_k$ applying the $2k-$th moment of the zeta. Building on the former paper, we apply an elementary approach (based on arithmetic averages) in order to get information on the reverse link to the second …

On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces

W. Luo has investigated the distribution of zeros of the derivative of the Selberg zeta function associated to compact hyperbolic Riemann surfaces. In essence, the main results in Luo's article involve the following three points: Finiteness for the number of zeros in the half plane to the left of the critical line; an asymptotic expansion for the counting function measuring the vertical distribution of zeros; and an asymptotic expansion for the counting function measuring the horizontal distance of zeros from the critical line. In the present article, we study the more complicated setting of d…

算术群 Selberg zeta 函数的普适性定理

Universality theorems of the Selberg zeta functions for arithmetic groups

本文将 Selberg zeta 函数的普适性定理从模群及其主同余子群进一步推广到模群的一般子群和由不定四元数代数导出的余紧算术群，并得到了比已有结果更宽的成立区域。

临界带中塞尔伯格zeta函数的欧拉乘积

Euler products of Selberg zeta functions in the critical strip

该研究扩展了模群同余子群对应塞尔伯格zeta函数欧拉乘积的收敛区域，建立了其与素测地线定理误差项的关系，在塞尔伯格猜想下可扩展至Re s≥3/4，对平凡表示则借鉴拉马努金工作思想证明。

A note on $r$-gaps between zeros of the Riemann zeta-function

In this note, we prove Selberg's announced result on $r$-gaps between zeros of the Riemann zeta-function $ζ$. Our proof uses a result on variations of $\argζ$ by Tsang based on Selberg's method. The same result with explicit constants under the Riemann Hypothesis has been obtained by Conrey and Turnage-Butterbaugh using a different method. We explain how to obtain explicit constants under the Riemann Hypothesis using our approach which is based on Selberg's and Tsang's arguments.

Computation of Selberg zeta functions on Hecke triangle groups

In this paper, a heuristic method to compute the Selberg zeta function for Hecke triangle groups, $G_q$ ($q>=3$) is described. The algorithm is based on the transfer operator method and an overview of the relevant background is given. We give numerical support for the claim that the method works and can be used to compute the Selberg Zeta function for $G_q$ to any desired precision. We also present some numerical results obtained by implementing the algorithm.

由 Zeta 零点线性统计量加权的 Selberg 中心极限定理

Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros

本文研究以 zeta 零点局部统计量加权的黎曼 zeta 函数对数值分布，证明其在 Fourier 支撑足够小时满足复中心极限定理，并在 Riemann 假设下将虚部支撑条件推广到自然障碍，同时刻画了 log ζ 与单层密度相关性的缓慢衰减。

Higher Selberg zeta functions for congruence subgroups

As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we determine the gamma factor explicitly in terms of the Barnes triple gamma function and the higher Dirichlet $L$-function.

Hierarchy of the Selberg zeta functions

We introduce a Selberg type zeta function of two variables which interpolates several higher Selberg zeta functions. The analytic continuation, the functional equation and the determinant expression of this function via the Laplacian on a Riemann surface are obtained.

An analogue of the Chowla-Selberg formula for several automorphic L-functions

In this paper, we will give a certain formula for the Riemann zeta function that expresses the Riemann zeta function by an infinte series consisting of $K$-Bessel functions. Such an infinite series expression can be regarded as an analogue of the Chowla-Selberg formula. Roughly speaking, the Chowla-Selberg formula is the formula that expresses the Epstein zeta-function by an infinite series consisting of $K$-Bessel functions. In addition, we also give certain analogues of the Chowla-Selberg formula for Dirichlet $L$-functions and $L$-functions associated with holomorphic cusp forms. Moreover, …

Superzeta functions, regularized products, and the Selberg zeta function on hyperbolic manifolds with cusps

Let $Λ= \{λ_{k}\}$ denote a sequence of complex numbers and assume that that the counting function $#\{λ_{k} \in Λ: | λ_{k}| < T\} =O(T^{n})$ for some integer $n$. From Hadamard's theorem, we can construct an entire function $f$ of order at most $n$ such that $Λ$ is the divisor $f$. In this article we prove, under reasonably general conditions, that the superzeta function $\Z_{f}(s,z)$ associated to $Λ$ admits a meromorphic continuation. Furthermore, we describe the relation between the regularized product of the sequence $z-Λ$ and the function $f$ as constructed as a Weierstrass product. In t…

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL(2,Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem, the asymptotic behavior of the sum of the class number.

Perturbation of zeros of the Selberg zeta-function for $Γ_0(4)$

We study the asymptotic behavior of zeros of the Selberg zeta-function for the congruence subgroup $Γ_0(4)$ as a function of a one-parameter family of characters tending to the trivial character. The motivation for the study comes from observations based on numerical computations. Some of the observed phenomena lead to precise theorems that we prove and compare with the original numerical results.

Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

We initiate the study of Selberg zeta functions $Z_{Γ,χ}$ for geometrically finite Fuchsian groups $Γ$ and finite-dimensional representations $χ$ with non-expanding cusp monodromy. We show that for all choices of $(Γ,χ)$, the Selberg zeta function $Z_{Γ,χ}$ converges on some half-plane in $\mathbb{C}$. In addition, under the assumption that $Γ$ admits a strict transfer operator approach, we show that $Z_{Γ,χ}$ extends meromorphically to all of $\mathbb{C}$.

Analogues of the Artin factorization formula for the automorphic scattering matrix and Selberg zeta-function associated to a Kleinian group

For Kleinian groups acting on hyperbolic three-space, we prove factorization formulas for both the Selberg zeta-function and the automorphic scattering matrix. We extend results of Venkov and Zograf from Fuchsian groups, to Kleinian groups, and we give a proof that is simple and extendable to more general groups.

Selberg's zeta functions for congruence subgroups of modular groups in SL(2,R) and SL(2,C)

It is known that the Selberg zeta function for the modular group has an expression in terms of the class numbers and the fundamental units of the indefinite binary quadratic forms. In the present paper, we generalize such a expression to any congruence subgroup of the modular groups in SL(2,R) and SL(2,C).

Remarks on a special value of the Selberg zeta function

Let $\CmZ_{Y_0(N)}$ be the constant term of the logarithmic derivative at $s=1$ of the Selberg zeta function of the modular curve $Y_0(N)$. Jorgenson and Kramer established the bound $\CmZ_{Y_0(N)}=O_ε(N^ε)$, $ε>0$ by relating it to geometric invariants. In this article we give, for $N$ prime, another proof via $L$-functions and exponential sums improving on a previous approach by Abbes-Ullmo and Michel-Ullmo. We further derive a power of $\log N$ bound along the same line.

Period functions for Hecke triangle groups, and the Selberg zeta function as a Fredholm determinant

We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions

Over the last few years Pohl (partly jointly with coauthors) developed dual `slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $Γ\backslash\mathbb{H}$ with cusps and all finite-dimensional unitary representations $χ$ of $Γ$. The eigenfunctions with eigenvalue $1$ of the fast transfer operators determine the zeros of the Selberg zeta function for $(Γ,χ)$. Further, if $Γ$ is cofinite and $χ$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue $1$ of the s…

Selberg's Conjectures and Artin $L$-functions

The author reviews results and conjectures of Selberg on a class of Dirichlet series functions which share properties with the Riemann zeta function, and he relates this work to the theory of Artin L-functions.

A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area

We provide an explicit construction of a cross section for the geodesic flow on infinite-area Hecke triangle surfaces which allows us to conduct a transfer operator approach to the Selberg zeta function. Further we construct closely related cross sections for the billiard flow on the associated triangle surfaces and endow the arising discrete dynamical systems and transfer operator families with two weight functions which presumably encode Dirichlet respectively Neumann boundary conditions. The Fredholm determinants of these transfer operator families constitute dynamical zeta functions, which…

Effective uniform approximation by $L$-functions in the Selberg class

Recently, Garunkštis, Laurinčikas, Matsumoto, J. & R. Steuding showed an effective universality-type theorem for the Riemann zeta-function by using an effective multi-dimensional denseness result of Voronin. We will generalize Voronin's effective result and their theorem to the elements of the Selberg class satisfying some conditions.

秩数为 Rankin-Selberg 赤字积分和分裂非温控 Gan-Gross-Prasad 猜想

Residues of Rankin-Selberg Zeta integrals and the split non-tempered Gan-Gross-Prasad conjectures

本文通过 Zeta 积分残差正则化方法，解决了非温控 Gan-Gross-Prasad 猜想及其细化版本在局部域上的问题。

Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary representations

Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,χ)$ for hyperbolic surfaces (orbifolds) $Γ\backslash\mathbb{H}$ as well as transfer operator techniques to develop period-like functions for $(Γ,χ)$-automorphic cusp forms. This leads to several natural conjectures. We further show how to extend these results to the billiard flow on the underlying triangle surfaces, and study the convergence of transfer operators along sequences of Hecke…

优化Selberg方法以研究临界零点

Optimising Selberg's method for critical zeros

本文优化了Selberg方法，通过改进算术平均值和引入半正定mollifiers，证明了Riemann zeta函数至少7%的非平凡零点位于临界线上，为该方法迄今最强结果。

Selberg Integrals, Multiple Zeta Values and Feynman Diagrams

We prove that there is an isomorphism between the Hopf Algebra of Feynman diagrams and the Hopf algebra corresponding to the Homogenous Multiple Zeta Value ring H in C<<X,Y>> . In other words, Feynman diagrams evaluate to Multiple Zeta Values in all cases. This proves a recent conjecture of Connes-Kreimer, and others including Broadhurst and Kontsevich. The key step of our theorem is to present the Selberg integral as discussed in Terasoma [22] as a Functional from the Rooted Trees Operad to the Hopf algebra of Multiple Zeta Values. This is a new construction which provides illumination to the…

Iterated integrals, multiple zeta values and Selberg integrals

Classical multiple zeta values can be viewed as iterated integrals of the differentials $\frac{dt}{t}, \frac{dt}{1-t}$ from $0$ to $1$. In this paper, we reprove Brown's theorem: For $a_i, b_i, c_{ij}\in \mathbb{Z}$, the iterated integral of the form \[ \mathop{\int\cdots \int}\limits_{0<t_1<\cdots<t_N<1}\prod_i t_i^{a_i}(1-t_i)^{b_i} \prod_{i<j}(t_j-t_i)^{c_{ij}}dt_1\cdots dt_N \] is a $\mathbb{Q}$-linear combination of multiple zeta values of weight $\leq N$ if convergent. What is more, we show that if $p_i(t), 1\leq i\leq N, $ are in a $\mathbb{Q}\left[t,1/t, 1/(1-t)\right]$-algebra generat…

Distribution of the zeta functions singularities for compact even-dimensional locally symmetric spaces

For compact, even-dimensional, locally symmetric spaces, we obtain precise estimates on the number of singularities of Selberg's and Ruelle's zeta functions considered by U. Bunke and M. Olbrich.

Order of zeta functions for compact odd-dimensional locally symmetric spaces

We prove that the meromorphic continuations of the Ruelle and Selberg zeta functions considered by Bunke and Olbrich are of finite order not larger than the dimension of the underlaying compact, odd-dimensional, locally symmetric space.

p-adic L-functions for Rankin-Selberg convolutions over number fields

We unconditionally construct cyclotomic p-adic L-functions for Rankin-Selberg convolutions for GL(n+1) x GL(n) over arbitrary number fields, and show that they satisfy an expected functional equation.

On the Rankin-Selberg problem: higher power moments of the Riesz mean error term

Let $Δ_1(x;ϕ)$ be the error term of the first Riesz means of the Rankin-Selberg zeta function. We study the higher power moments of $Δ_1(x;ϕ)$ and derive an asymptotic formula for 3-rd, 4-th and 5-th power moments by using Ivić 's large value arguments.

A note on p-adic Rankin--Selberg L-functions

We prove an interpolation formula for the values of certain $p$-adic Rankin--Selberg $L$-functions associated to non-ordinary modular forms.

The universality of symmetric power L-functions and their Rankin-Selberg L-functions

We establish the universality theorem for the first four symmetric power L-functions of automorphic forms and their associated Rankin-Selberg L-functions. This generalizes some results of Laurincikas & Matsumoto and Matsumoto respectively.

On the logarithmic derivative of zeta functions for compact even-dimensional locally symmetric spaces

We derive approximate formulas for the logarithmic de- rivative of the Selberg and Ruelle zeta functions over compact, even- dimensional, locally symmetric spaces of rank one. The obtained for- mulas are given in terms of the zeta-singularities.

On the Vertical Distribution of Values of $L$-functions in the Selberg Class

We prove explicit formulae for $α$-points of $L$-functions from the Selberg class. Next we extend a theorem of Littlewood on the vertical distribution of zeros of the Riemann zeta-function $ζ(s)$ to the case of $α$-points of the aforementioned $L$-functions. This result implies the uniform distribution of subsequences of $α$-points and from this a discrete universality theorem in the spirit of Voronin is derived.

Real zeros and size of Rankin-Selberg L-functions in the level aspect

In this paper, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg L-functions. One of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. A first consequence is a new subconvexity bound for Rankin-Selberg L-functions in the level aspect. Moreover, infinitely many Rankin-Selberg L-functions having at most eight non-trivial real zeros are produced and some new non-trivial estimates for the analytic rank of the family…

Subconvexity for the Rankin-Selberg L-function in both levels

In this paper, we obtain a subconvexity result for the Rankin-Selberg L-function in both levels. The new feature in this result is applying an amplification method of Duke-Friedlander-Iwaniec to a double Petersson-Kuznetsov trace formula. As the trace formula ranges over all the $GL_2$ spectrum, this subconvexity result is unconditional of which Hecke eigenforms are chosen in the Rankin-Selberg L-function.

GL₂×GL₂秩纳积分的积分性：对于分裂表示

Integrality of $\mathrm{GL}_2\times\mathrm{GL}_2$ Rankin-Selberg integrals for ramified representations

本文研究GL₂×GL₂秩纳积分在分裂表示下的积分性，引入了(π₁×π₂)积分数据的概念，并改进了雅克布-兰格兰德的GCD结果，与新形式的傅里叶系数的积分性相容。

New integral representations for Rankin-Selberg L-functions

We derive integral representations for the Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2) by a process of unipotent averaging at archimedean places. A key feature of our result is that it allows one to fix the choice of test vector at finite places, irrespective of ramification. This enables a new proof of the functional equation for GL(3) x GL(2) Rankin-Selberg L-functions in many cases.

Beyond endoscopy for the Rankin-Selberg L-function

We try to understand the poles of L-functions via taking a limit in a trace formula. This technique avoids endoscopic and Kim-Shahidi methods. In particular, we investigate the poles of the Rankin-Selberg L-function. Using analytic number theory techniques to take this limit, we essentially get a new proof of the analyticity of the Rankin-Selberg L-function at $s=1.$ Along the way we discover the convolution operation for Bessel transforms.

A standard zero free region for Rankin-Selberg L-functions

A standard zero free region is obtained for Rankin Selberg L-functions $L(s, f\times \widetilde{f})$ where $f$ is an almost everywhere tempered Maass form on $GL(n)$ and $f$ is not necessarily self dual. The method is based on the theory of Eisenstein series generalizing a work of Sarnak.

On the Rankin--Selberg problem

In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.

On a Rankin-Selberg integral of the L-function for $\widetilde{SL}_2\times GL_2$

We present a Rankin-Selberg integral on the exceptional group $G_2$ which represents the L-function for generic cuspidal representations of $\widetilde{SL}_2\times GL_2$. As an application, we show that certain Fourier-Jacobi type periods on $G_2$ are non-vanishing.