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

TL;DR

This paper provides explicit unconditional bounds on the argument of Dirichlet L-functions and estimates the height of their lowest non-trivial zeros, with applications to understanding low-lying zeros.

Contribution

It makes explicit a classical result of Selberg and derives new bounds on the height of the lowest zeros of Dirichlet L-functions for large primes.

Findings

Height of the lowest zero is less than 1075 times the average zero spacing for large primes q.

Proportion of L-functions with first zero within a multiple of average spacing is bounded below.

First explicit unconditional results on low-lying zeros of Dirichlet L-functions.

Abstract

We make explicit a result of Selberg on the argument of Dirichlet $L$-functions averaged over non-principal characters modulo a prime $q$. As a corollary, we show for all sufficiently large prime $q$ that the height of the lowest non-trivial zero of the corresponding family of $L$-functions is less than $1075\cdot \frac{2\pi}{\log q}$. Here the scaling factor $\frac{2\pi}{\log q}$ is the average spacing between consecutive low-lying zeros with height at most 1, say. We also obtain a lower bound on the proportion of $L$-functions whose first zero lies within a given multiple of the average spacing. These appear to be the first explicit unconditional results of their kinds.