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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

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).

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

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