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

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 $\hat{I}›(s,\ddot{I}{‡}_D)$ at the central point $s=1/2$. In particular, we show that the proportion of $\hat{I}{›}^{(2k)}(\frac{1}{2},\ddot{I}{‡}_D)\ne 0$ is $1+O(k^{-2})$ as $k\to \infty$.

20 pages, significantly revised version