Related papers: On primeness of the Selberg zeta-function

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…

For a compact Riemann surface $M$ of genus $g\ge 2$, we study the functional equations of the Selberg zeta functions attached with the Tate motives $f$. We prove that certain functional equations hold if and only if $f$ has the absolute…

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely…

In this note we show the equivalence of Benjamini-Schramm convergence and convergence of the zeta functions for compact hyperbolic surfaces.

In this paper, we study the Selberg and Ruelle zeta functions on compact hyperbolic odd dimensional manifolds. These zeta functions are defined on one complex variable $s$ in some right half-plane of $\mathbb{C}$. We use the Selberg trace…

This paper is the third in a sequel to develop a super-analogue of the classical Selberg trace formula, the Selberg supertrace formula. It deals with bordered super Riemann surfaces. The theory of bordered super Riemann surfaces is…

It is shown that the zeta functions of Ruelle and Selberg admit analytic continuation to meromorphic functions on the plane for every compact locally-symmetric space and every non-unitary twist.

Theta functions play a major role in many current researches and are powerful tools for studying integrable systems. The purpose of this paper is to provide a short and quick exposition of some aspects of meromorphic theta functions for…

We collect some classical results related to analysis on the Riemann surfaces. The notes may serve as an introduction to the field: we suppose that the reader is familiar only with the basic facts from topology and complex analysis. the…

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…

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…

We present the first example of the Selberg type zeta function for noncompact higher rank locally symmetric spaces. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real…

In this work we show that the Riemann hypothesis for the Dedekind zeta--function $\zeta_{\mathrm{K}}(s)$ of an algebraic number field $\mathrm{K}$ is equivalent to a problem of the rate of convergence of certain discrete measures defined…

Using elementary methods we find surprising connections between the values of the Riemann Zeta Function over integers and the fractional parts of rational powers, and a connection between the Riemann Zeta Function and the Prime Zeta…

In this short note, we give a proof of the Riemann hypothesis for Goss $v$-adic zeta function $\zeta_{v}(s)$, when $v$ is a prime of $\mathbb{F}_{q}[t]$ of degree one.

We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable $s$ in some right-half…

By considering the prime zeta function, the author intended to demonstrate in that the Riemann zeta function zeta(s) does not vanish for Re(s)>1/2, which would have proven the Riemann hypothesis. However, he later realised that the proof of…

This paper continues a series of investigations on converging representations for the Riemann Zeta function. We generalize some identities which involve Riemann's zeta function, and moreover we give new series and integrals for the zeta…

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…

The purpose of this paper is to prove that the so-called Quasi-Riemann Hypothesis for the Zeta-function implies the Riemann Hypothesis