One-level density estimates for Dirichlet L-functions with extended support

Drappeau, Sary; Pratt, Kyle; Radziwiłł, Maksym

doi:10.2140/ant.2023.17.805

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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

DOI: 10.2140/ant.2023.17.805

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 $[\frac{1}{2} Q, Q]$ and for test functions whose Fourier transform is supported in $(- 2 - \frac{50}{1093}, 2 + \frac{50}{1093})$ . 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

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Vol. 17, No. 4, 2023