The Epstein zeta-function contains a positive proportion of non-trivial zeros on the critical line

TL;DR

This paper proves that the Epstein zeta-function for certain quadratic forms has a positive proportion of its non-trivial zeros on the critical line, advancing understanding of its zeros' distribution.

Contribution

It establishes that the Epstein zeta-function associated with binary positive definite quadratic forms has a positive proportion of zeros on the critical line, a new result in analytic number theory.

Findings

Positive proportion of zeros on the critical line

Advances understanding of zeros distribution in Epstein zeta-functions

Extends results to quadratic forms with integer coefficients

Abstract

It is proved that the Epstein zeta-function corresponding to a binary positive definite quadratic form with integer coefficients has a positive proportion of its non-trivial zeros on the critical line.