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

Volume 18
Issue 6, 1039–1219
Issue 5, 847–1038
Issue 4, 631–846
Issue 3, 409–629
Issue 2, 209–408
Issue 1, 1–208

Volume 17, 12 issues

Volume 16, 10 issues

Volume 15, 10 issues

Volume 14, 10 issues

Volume 13, 10 issues

Volume 12, 10 issues

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
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
One-level density estimates for Dirichlet $L$-functions with extended support

Sary Drappeau, Kyle Pratt and Maksym Radziwiłł

Vol. 17 (2023), No. 4, 805–830
Abstract

We estimate the 1-level density of low-lying zeros of L(s,χ) with χ ranging over primitive Dirichlet characters of conductor in [1 2Q,Q] and for test functions whose Fourier transform is supported in ( 2 50 1093,2 + 50 1093). Previously, any extension of the support past the range (2,2) was only known conditionally on deep conjectures about the distribution of primes in arithmetic progressions, beyond the reach of the generalized Riemann hypothesis (e.g., Montgomery’s conjecture). Our work provides the first example of a family of L-functions in which the support is unconditionally extended past the “diagonal range” that follows from a straightforward application of the underlying trace formula (in this case orthogonality of characters). We also highlight consequences for nonvanishing of L(s,χ).

Keywords
Dirichlet $L$-functions, one-level density, nonvanishing, primes, arithmetic progressions, dispersion method
Mathematical Subject Classification
Primary: 11M26
Secondary: 11M50, 11N13
Milestones
Received: 28 April 2020
Revised: 31 January 2022
Accepted: 10 June 2022
Published: 2 May 2023
Authors
Sary Drappeau
Institut de Mathématiques de Marseille
Aix-Marseille Université
CNRS
Marseille
France
Kyle Pratt
Mathematical Institute
All Souls College
University of Oxford
Oxford
United Kingdom
Maksym Radziwiłł
Division of Physics, Mathematics and Astronomy
Caltech
Pasadena, CA
United States

Open Access made possible by participating institutions via Subscribe to Open.