Related papers: The Lerch Zeta Function I. Zeta Integrals

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with…

Observing a multiple version of the divisor function we introduce a new zeta function which we call a multiple finite Riemann zeta function. We utilize some $q$-series identity for proving the zeta function has an Euler product and then,…

This is primarily an overview article on some results and problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it){\bigl(\chi(1/2+it)\bigr)}^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s). $$ In particular, we discuss the first…

We introduce an "$L$-function" $\mathcal{L}$ built up from the integral representation of the Barnes' multiple zeta function $\zeta$. Unlike the latter, $\mathcal{L}$ is defined on a domain equipped with a non-trivial action of a group $G$.…

This article introduces and investigates the basic features of a dynamical zeta function for group actions, motivated by the classical dynamical zeta function of a single transformation. A product formula for the dynamical zeta function is…

This paper establishes new bridges between number theory and modern harmonic analysis, namely between the class of complex functions, which contains zeta functions of arithmetic schemes and closed with respect to product and quotient, and…

We consider Cauchy type integrals $I(t)={1\over 2\pi i}\int_{\gamma} {g(z)dz\over z-t}$ with $g(z)$ an algebraic function. The main goal is to give constructive (at least, in principle) conditions for $I(t)$ to be an algebraic function, a…

This paper introduces a set of finite summation formulas and utilize them to establish various functional relationships involving the multivariable Hurwitz-Lerch zeta function. Additionally, the paper examines several examples of these…

We study some classical identities for multiple zeta values and show that they still hold for zeta functions built on the zeros of an arbitrary function. We introduce the complementary zeta function of a system, which naturally occurs when…

In this paper, we present two new representations of the alternating Zeta function. We show that for any s $\in$ C this function can be computed as a limit of a series of determinant. We then express these determinants as the expectation of…

We construct variants of the Riemann zeta function with convenient properties and make conjectures about their dynamics; some of the conjectures are based on an analogy with the dynamical system of zeta. More specifically, we study the…

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane -- that is, a nondegenerate isotropic quadratic space of dimension $2$. We…

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…

A formula for the Hurwitz zeta function at the positive integers $k$, $\zeta(k,b)$, is created by solving the real and the imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known…

We determine the special values at positive integers of the spectral zeta function associated with the combinatorial Laplacian on the regular tree. These values admit explicit formulas in terms of certain polynomials, which we show to be…

We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an…

We define reduced zeta functions of Lie algebras, which can be derived from motivic zeta functions using the Euler characteristic. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to…

We, by making use of elementary arguments, deduce integral representations of the Legendre chi function $\chi_{s}(x)$ valid for $|z|<1$ and $\Re(s)>1$. Our earlier established results on the integral representations for the Riemann zeta…

We study relationships between spinor representations of certain Lie algebras and Lie superalgebras of differential operators on the circle and values of $\zeta$--functions at the negative integers. By using formal calculus techniques we…

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of…