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One-level density
estimates for Dirichlet $L$-functions with extended
support
Sary Drappeau, Kyle Pratt and Maksym Radziwiłł |
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Vol. 17 (2023), No. 4, 805–830
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Abstract |
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We estimate the -level density of low-lying zeros of with ranging over primitive Dirichlet characters of conductor in and for test functions whose Fourier transform is supported in . Previously, any extension of the support past the range 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 -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 . |
Keywords
Dirichlet $L$-functions, one-level density, nonvanishing,
primes, arithmetic progressions, dispersion method
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Mathematical Subject Classification
Primary: 11M26
Secondary: 11M50, 11N13
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Milestones
Received: 28 April 2020
Revised: 31 January 2022
Accepted: 10 June 2022
Published: 2 May 2023
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Authors |
| Sary Drappeau | |
| Institut de Mathématiques de
Marseille Aix-Marseille Université CNRS Marseille France |
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| Kyle Pratt | |
| Mathematical Institute All Souls College University of Oxford Oxford United Kingdom |
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| Maksym Radziwiłł | |
| Division of Physics, Mathematics and
Astronomy Caltech Pasadena, CA United States |
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| © 2023 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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