Average nonvanishing of Dirichlet $L$-functions at the central point

Abstract

The generalized Riemann hypothesis implies that at least 50% of the central values $L\left(\frac{1}{2},\chi \right)$ are nonvanishing as $\chi$ ranges over primitive characters modulo $q$. We show that one may unconditionally go beyond GRH, in the sense that if one averages over primitive characters modulo $q$ and averages $q$ over an interval, then at least 50.073% of the central values are nonvanishing. The proof utilizes the mollification method with a three-piece mollifier, and relies on estimates for sums of Kloosterman sums due to Deshouillers and Iwaniec.