Non-Vanishing of Dirichlet $L$-functions at the central point with restricted root number

TL;DR

This paper establishes that a positive proportion of Dirichlet L-functions with prime conductor do not vanish at the central point, using mollified moments and angular restrictions on root numbers.

Contribution

It introduces a new method to prove non-vanishing of Dirichlet L-functions at the central point under angular restrictions on root numbers.

Findings

Positive proportion of non-vanishing L-values for prime conductors

Asymptotic formulas for mollified moments under restrictions

Extension of non-vanishing results to restricted root number subfamilies

Abstract

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime conductor $q$, a positive proportion of the central values $L(1/2,\chi)$ do not vanish as $q\to\infty$.