Vol. 13, No. 1, 2019

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

Kyle Pratt

Vol. 13 (2019), No. 1, 227–249
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.

Keywords
Dirichlet $L$-function, nonvanishing, central point, mollifier, sums of Kloosterman sums
Primary: 11M06
Secondary: 11M26