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

TL;DR

This paper establishes bounds on the density of zeros near the critical line for Dirichlet L-functions, providing new asymptotic formulas and demonstrating that at least 38.2% of zeros lie on the critical line.

Contribution

It introduces an asymptotic for the twisted second moment of Dirichlet L-functions and applies it to show a lower bound on zeros on the critical line for all moduli q.

Findings

At least 38.2% of zeros lie on the critical line for any modulus q.

Derived an asymptotic formula for the twisted second moment of Dirichlet L-functions.

Provided an upper bound on the density of zeros close to the critical line.

Abstract

We prove an upper bound on the density of zeros very close to the critical line of the family of Dirichlet $L$-functions of modulus $q$ at height $T$. To do this, we derive an asymptotic for the twisted second moment of Dirichlet $L$-functions uniformly in $q$ and $t$. As a second application of the asymptotic formula we prove that, for every integer $q$, at least $38.2\%$ of zeros of the primitive Dirichlet $L$-functions of modulus $q$ lie on the critical line.