Related papers: Selberg, Ihara and Berkovich

In this paper, we establish Kronecker limit type formulas for the Mordell-Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the second as well as the third arguments. As an application of these formulas, we obtain results of…

For certain spectral parameters we find explicit eigenfunctions of transfer operators for Schottky surfaces. Comparing the dimension of the eigenspace for the spectral parameter zero with the multiplicity of topological zeros of the Selberg…

For any congruence subgroup of the modular group, we extend the region of convergence of the Euler products of the Selberg zeta functions beyond the boundary Re s = 1, if they are attached with a nontrivial irreducible unitary…

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…

We introduce a deficiency-based representation and approximation framework for values of the Riemann zeta function. The method is based on comparing two nonlinear accumulation mechanisms: global transformation of a base partial sum and…

We study the asymptotic behaviour of Bessel functions associated of root systems of type $A_{n-1}$ and type $B_n$ with positive multiplicities as the rank $n$ tends to infinity. In both cases, we characterize the possible limit functions…

We study the \lq \lq symmetry integral\rq \rq, \thinspace say $I_f$, of some arithmetic functions $f:\N \rightarrow \R$; we obtain from lower bounds of $I_f$ (for a large class of arithmetic functions $f$) lower bounds for the \lq \lq…

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…

We describe an efficient iterative algorithm for the computation of theta functions of non-archimedean Schottky groups and, more generally, of (non-archimedean) discontinuous groups.

We study analytic properties of the representation zeta functions of arithmetic groups of type $\mathsf{A}_2$, such as $\textrm{SL}_3(\mathbb{Z})$. In particular, we uncover further poles of these functions and determine a natural boundary…

We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry…

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.

Let $\Gamma_{w}$ be a non-cofinite Hecke triangle group with cusp width $w>2$ and let $\varrho\colon\Gamma_w\to U(V)$ be a finite-dimensional unitary representation of $\Gamma_w$. In this note we announce a new fractal upper bound for the…

We will investigate the relationship between Ihara's zeta functions of Ramanujan graphs and Hasse-Weil's congruent zeta functions of modular curves. As an application we will describe the limit value of Hasse-Weil's congruent zeta functions…

We introduce the zeta function of the prehomogenous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues.…

By a similar idea for the construction of Milnor's gamma functions, we introduce "higher depth determinants" of the Laplacian on a compact Riemann surface of genus greater than one. We prove that, as a generalization of the determinant…

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and…

In this paper, we define edge zeta functions for spherical buildings associated with finite general linear groups. We derive elegant formulas for these zeta functions and reveal patterns of eigenvalues of these buildings, by introducing and…

We find an asymptotic expansion of a multi-dimensional version of Selberg's central limit theorem for $L$-functions on $ \sigma= \frac12 + ( \log T)^{-\theta}$ and $ t \in [ T, 2T]$, where $ 0 < \theta < \frac12 $ is a constant.

We revisit congruence zeta functions of smooth projective varieties over finite fields in the framework of Scholze's Berkovich motives. Via this formalism and categorical traces, we construct a new zeta function, and show that it agree with…