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Low-lying zeros of quadratic Dirichlet $L$-functions: lower order terms for extended support

Part of: Exponential sums and character sums Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  26 April 2017

Daniel Fiorilli ,
James Parks  and
Anders Södergren
Daniel Fiorilli
Affiliation:
Département de mathématiques et de statistique, Université d’Ottawa, 585 King Edward, Ottawa, ON K1N 6N5, Canada email [email protected]
James Parks
Affiliation:
Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive, Lethbridge, AB T1K 3M4, Canada email [email protected] Current address: Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen, SE-100 44 Stockholm, Sweden
Anders Södergren
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark email [email protected] Current address: Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Gothenburg, Sweden
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Abstract

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions attached to real primitive characters of conductor at most $X$. Under the generalized Riemann hypothesis, we give an asymptotic expansion of this quantity in descending powers of $\log X$, which is valid when the support of the Fourier transform of the corresponding even test function $\unicode[STIX]{x1D719}$ is contained in $(-2,2)$. We uncover a phase transition when the supremum $\unicode[STIX]{x1D70E}$ of the support of $\widehat{\unicode[STIX]{x1D719}}$ reaches $1$, both in the main term and in the lower order terms. A new lower order term appearing at $\unicode[STIX]{x1D70E}=1$ involves the quantity $\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.

Keywords

zeros of L-functionsKatz–Sarnak heuristicsquadratic Dirichlet L-functions1-level density

MSC classification

Primary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M50: Relations with random matrices
Secondary: 11L40: Estimates on character sums
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 6 , June 2017 , pp. 1196 - 1216
Copyright
© The Authors 2017 

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