Mollified moments of quadratic Dirichlet $L$-functions over function fields

TL;DR

This paper derives asymptotic formulas for mollified moments of quadratic Dirichlet L-functions over function fields, leading to non-vanishing results for derivatives at the central point.

Contribution

It provides the first asymptotic formulas for mollified moments in this setting and establishes non-vanishing proportions for derivatives of L-functions at s=1/2.

Findings

Asymptotic formulas for mollified first and second moments.

Non-vanishing of derivatives of L-functions at the central point.

Proportion of non-vanishing derivatives approaches 1 as order increases.

Abstract

We compute asymptotic formulae for the mollified first and second moments for the family of quadratic Dirichlet $L$-functions in the function field setting. As an application, we obtain non-vanishing results for the derivatives of the completed $L$-functions $\Lambda(s,\chi_D)$ at the central point $s=1/2$. In particular, we show that the proportion of $\Lambda^{(2k)}(\frac{1}{2},\chi_D) \neq 0$ is $1+O(k^{-2})$ as $k \to \infty$.