Distinct zeros and simple zeros of Dirichlet $L$-functions

TL;DR

This paper investigates the multiplicity of zeros of Dirichlet L-functions, establishing that a majority are simple and distinct, with improved proportions under the Generalized Riemann Hypothesis using the Asymptotic Large Sieve method.

Contribution

It provides new asymptotic estimates for the proportions of simple and distinct zeros of Dirichlet L-functions, including improvements assuming the GRH.

Findings

Over 80% of zeros are distinct

Over 60% of zeros are simple

Proportions increase under GRH

Abstract

In this paper, we study the number of additional zeros of Dirichlet $L$-function caused by multiplicity by using Asymptotic Large Sieve. Then in asymptotic terms we prove that there are more than 80.124% of zeros of the family of Dirichlet $L$-functions are distinct and more than 60.248% of zeros of the family of Dirichlet $L$-functions are simple. In addition, assuming the Generalized Riemann Hypothesis, we improve these proportions to 83.216% and 66.433%.