On the zeros of the Epstein zeta function

TL;DR

This paper investigates the distribution of zeros of the Epstein zeta function associated with a quadratic form, focusing on the count of consecutive zeros on the critical line within a large interval separated by a fixed positive distance.

Contribution

It provides a new method to count consecutive zeros of the Epstein zeta function on the critical line with specified spacing, advancing understanding of its zero distribution.

Findings

Count of consecutive zeros in large intervals

Zeros separated by a positive number V

Distribution pattern of zeros on the critical line

Abstract

In this article, we count the number of consecutive zeros of the Epstein zeta-function, associated to a certain quadratic form, on the critical line with ordinates lying in $[0,T], T$ sufficiently large and which are separated apart by a given positive number $V$.