Average nonvanishing of Dirichlet L-functions at the central point

Pratt, Kyle

doi:10.2140/ant.2019.13.227

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

Kyle Pratt

Vol. 13 (2019), No. 1, 227–249

DOI: 10.2140/ant.2019.13.227

Abstract

The generalized Riemann hypothesis implies that at least 50% of the central values $L (\frac{1}{2}, χ)$ are nonvanishing as $χ$ 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.

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Keywords

Dirichlet $L$-function, nonvanishing, central point, mollifier, sums of Kloosterman sums

Mathematical Subject Classification 2010

Primary: 11M06

Secondary: 11M26

Milestones

Received: 4 April 2018

Revised: 23 July 2018

Accepted: 23 September 2018

Published: 13 February 2019

Authors

Kyle Pratt
Department of Mathematics University of Illinois at Urbana–Champaign Urbana, IL United States

Vol. 13, No. 1, 2019