Conditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros

TL;DR

This paper employs Beurling-Selberg extremal functions under GRH to bound the argument of Dirichlet L-functions, leading to new insights on low-lying zeros and their distribution.

Contribution

It provides alternative proofs for results on low-lying zeros and establishes a new lower bound on the proportion of L-functions with zeros near the central point.

Findings

Conditional existence of many L-functions with zeros below a certain height

New lower bound on the proportion of zeros close to the central point

Improved understanding of zero distribution under GRH

Abstract

Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet $L$-functions to a large prime modulus $q$. As applications, we give alternative proofs of several results on low-lying zeros of $L(s,\chi)$ and obtain a new lower bound on the proportion of $L(s,\chi)$ modulo $q$ with zeros close to the central point $s=1/2$. In particular, we show conditionally that for any $\beta>1/4$, there exist a positive proportion of Dirichlet $L$-functions whose first zero has height less than $\beta$ times the average spacing between consecutive zeros.