Vol. 13, No. 1, 2019

Download this article
Download this article For screen
For printing
Recent Issues

Volume 12
Issue 10, 2237–2514
Issue 9, 2033–2235
Issue 8, 1823–2032
Issue 7, 1559–1821
Issue 6, 1311–1557
Issue 5, 1001–1309
Issue 4, 751–999
Issue 3, 493–750
Issue 2, 227–492
Issue 1, 1–225

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors' Addresses
Editors' Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Other MSP Journals
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