Related papers: Two-variable zeta functions and regularized produc…

The main goal of this paper is to prove a formula that expresses the limit behaviour of Dedekind zeta functions for $\Re s > 1/2$ in families of number fields, assuming that the Generalized Riemann Hypothesis holds. This result can be…

We introduce new non-abelian zeta functions for curves defined over finite fields. There are two types, i.e., pure non-abelian zetas defined using semi-stable bundles, and group zetas defined for pairs consisting of (reductive group,…

We introduce and survey results on two families of zeta functions connected to the multiplicative and additive theories of integer partitions. In the case of the multiplicative theory, we provide specialization formulas and results on the…

This is the second of two papers introducing and investigating two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. In the first part, we proved some of their properties such as…

We define the doubling zeta integral for smooth families of representations of classical groups. Following this we prove a rationality result for these zeta integrals and show that they satisfy a functional equation. Moreover, we show that…

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard…

It is known that there are infinitely many singularities of multiple zeta functions and the special values at non-positive integer points are indeterminate. In order to give a suitable rigorous meaning of the special values there, Furusho,…

We compute the local pro-isomorphic zeta functions at all but finitely many primes for a certain family of class-two-nilpotent Lie lattices of even rank, parametrized by irreducible non-linear polynomials $f(x) \in \mathbb{Z} [x]$, that…

In this contribution we announce a complete classification and new exotic phenomena of the meromorphic structure of $\z$-functions associated to conic manifolds proved in \cite{KLP1}. In particular, we show that the meromorphic extensions…

In this article, we develop a k-free zeta Dirichlet series into a Laurent series with a simple pole, and prove a Stieltjes like formula for the expansion coefficients of the regular part. We also investigate another analytical continuation…

The two-dimensional inhomogeneous zeta-function series (with homogeneous part of the most general Epstein type): \[ \sum_{m,n \in \mbox{\bf Z}} (am^2+bmn+cn^2+q)^{-s}, \] is analytically continued in the variable $s$ by using zeta-function…

An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.

In a recent paper Kachi and Tzermias give elementary proofs of four product formulas involving zeta(3), pi, and Catalan's constant. They indicate that they were not able to deduce these products directly from the values of a function…

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings and…

In this paper, we generalize the partial fraction decomposition which is fundamental in the theory of multiple zeta values, and prove a relation between Tornheim's double zeta functions of three complex variables. As applications, we give…

Using the fact that a finite sum of power series are given by the difference between two zeta functions, we justify the usage of the zeta function with a negative variable in physical problems to avoid the divergence of the infinite sum. We…

In 1966, Tate proposed the Artin--Tate conjectures, which expresses special values of zeta function associated to surfaces over finite fields. Conditional on the Tate conjecture, Milne--Ramachandran formulated and proved similar conjectures…

In this paper, we consider meromorphic extension of the function \[ \zeta_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call…

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…

We introduce and study the mixed Segre zeta function of a sequence of homogeneous ideals in a polynomial ring. This function is a power series encoding information about the mixed Segre classes obtained by extending the ideals to projective…