Related papers: A functional equation and degenerate principal ser…

We show how to obtain the difference function of the Weierstrass Zeta function very directly, by choosing an appropriate order of summation for the series defining this function. As a byproduct, we show how to obtain the quasi-modularity of…

Using analytic torsion associated to stable bundles, we introduce zeta functions for compact Riemann surfaces. To justify the well-definedness, we analyze the degenerations of analytic torsions at the boundaries of the moduli spaces, the…

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete…

The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which…

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…

This paper develops a generalized cotangent-type series, extending classical expansions to higher-order lattice sums. By introducing a new family of series indexed by integer powers, we derive closed form representations that combine…

Starting from a recent result expressing the Lerch zeta function as a fractional derivative, we consider further fractional derivatives of the Lerch zeta function with respect to different variables. We establish a partial differential…

We introduce a new type of multiple zeta functions, which we call bilateral zeta functions, analogous to the Barnes zeta functions. The bilateral zeta function is a periodic function and shares certain basic properties of Barnes zeta…

We prove or conjecture several relations between the multizeta values for positive genus function fields of class number one, focusing on the zeta-like values, namely those whose ratio with the zeta value of the same weight is rational (or…

As a function of second variable, we identify the Fourier series of Hurwitz zeta function and its derivatives on the unit interval. Consequently, we obtain results based on the formula for Fourier coefficients and also on Parseval's…

We derive and prove a new formulation of the Lerch zeta function as a fractional derivative of an elementary function. We demonstrate how this formulation interacts very naturally with basic known properties of Lerch zeta, and use the…

We prove a general result on representing the Riemann zeta function as a convergent infinite series in a complex vertical strip containing the critical line. We use this result to re-derive known expansions as well as to discover new series…

Throughout this paper, k is a number field. We fix a quadratic extension k_1=k(a_0) of k, where a_0=sqrt(B_0) for a certain B_0 in k^\times/(k^\times)^2. In this paper, we consider the zeta function defined for the space of pairs of binary…

A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this…

A simple and elementary derivation of values at integer points for the Riemann's zeta and related functions is reported.

We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These…

Using the cohomology theory of Dwork, as developed by Adolphson and Sperber, we exhibit a deterministic algorithm to compute the zeta function of a nondegenerate hypersurface defined over a finite field. This algorithm is particularly…

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…

A possible connection between quantum computing and Zeta functions of finite field equations is described. Inspired by the 'spectral approach' to the Riemann conjecture, the assumption is that the zeroes of such Zeta functions correspond to…

We have dealt with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and showed some evidence to indicate the hypothesis. We briefly…