On the Epstein zeta function and the zeros of a class of Dirichlet series

TL;DR

This paper generalizes the Selberg-Chowla formula to establish analytic continuation and functional equations for a broad class of Epstein zeta functions, enabling new insights into the zeros of related Dirichlet series.

Contribution

It introduces a new method for analyzing zeros of Dirichlet series by extending classical formulas and exploring symmetries in Epstein zeta functions.

Findings

Established analytic continuation and functional equations for new Epstein zeta functions

Derived generalizations of classical formulas in analytic number theory

Provided a new approach to study zeros of Dirichlet series

Abstract

By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems regarding the distribution of zeros of Dirichlet series in their critical lines and to produce a new method for the study of these problems. Due to the symmetries provided by the representation via the Selberg-Chowla formula, some generalizations of well-known formulas in analytic number theory are also deduced as examples.