On the zeros of Epstein zeta functions

TL;DR

This paper studies the distribution of zeros of Epstein zeta functions related to positive definite quadratic forms with rational coefficients, providing asymptotic formulas that improve previous bounds on the number of zeros off the critical line.

Contribution

It offers new asymptotic formulas for counting zeros of Epstein zeta functions, advancing understanding of their distribution beyond prior lower bounds.

Findings

Derived asymptotic formulas for zero counts

Improved bounds on zeros off the critical line

Enhanced understanding of zeros distribution in Epstein zeta functions

Abstract

We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients. Davenport and Heilbronn, and also Voronin, proved the existence of zeros of Epstein zeta functions off the critical line when the class number of the quadratic form is bigger than 1. These authors give lower bounds for the number of zeros in strips that are of the same order as the more easily proved upper bounds. In this paper, we improve their results by providing asymptotic formulas for the number of zeros.