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
DOI: 10.2140/ant.2019.13.227
Abstract

The generalized Riemann hypothesis implies that at least 50% of the central values L(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.

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