Zeros of Dirichlet $L$-functions on the critical line

TL;DR

This paper estimates the proportion of zeros of Dirichlet L-functions on the critical line, showing over 60% are on the line and are simple, using advanced mollifier techniques and asymptotic formulas.

Contribution

It provides improved lower bounds for the proportion of zeros on the critical line of Dirichlet L-functions using Feng's mollifier and mean square estimates.

Findings

At least 61.07% of zeros are on the critical line.

At least 60.44% of zeros are simple and on the critical line.

Results improve previous bounds by Conrey, Iwaniec, and Soundararajan.

Abstract

In this paper, we estimate the proportion of zeros of Dirichlet $L$-functions on the critical line. Using Feng's mollifier and an asymptotic formula for the mean square of Dirichlet $L$-functions, we prove that averaged over primitive characters and conductors, at least 61.07 % of zeros of Dirichlet $L$-functions are on the critical line, and at least 60.44 % of zeros are simple and on the critical line. These results improve the work of Conrey, Iwaniec and Soundararajan.