Average non-vanishing 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 non-vanishing 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 non-vanishing. 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. Note: The author has been made aware of an error in this work. It seems the error can be fixed, by using a different argument, and the author will present a correction in due course.

22 pages. Added notice of an error in the work. Corrected proof to appear later

References in corpus (1)

• Nonvanishing of Dirichlet L-functions

Cited by in corpus (1)

• One-level density estimates for Dirichlet L-functions with extended support