Zero-density estimates for Epstein zeta functions

TL;DR

This paper studies the zeros of Epstein zeta functions for quadratic forms with rational coefficients within a specific vertical strip, improving bounds on the error term for the zero count when the class number exceeds one.

Contribution

It provides a new upper bound for the error term in the zero counting function of Epstein zeta functions, refining previous results by Voronin and Lee.

Findings

Improved upper bound for the error term in zero counting

Enhanced understanding of zero distribution for quadratic forms with class number > 1

Refined asymptotic estimates for zeros in a vertical strip

Abstract

We investigate the zeros of Epstein zeta functions associated with a positive definite quadratic form with rational coefficients in the vertical strip $ \sigma_1 < \Re s < \sigma_2 $, where $ 1/2 < \sigma_1 < \sigma_2 < 1 $. When the class number of the quadratic form is bigger than 1, Voronin gives a lower bound and Lee gives an asymptotic formula for the number of zeros. In this paper, we improve their results by providing a new upper bound for the error term.