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

Journal article
Authors D. Fiorilli
J. Parks
Anders Södergren
Published in Compositio Mathematica
Volume 153
Issue 6
Pages 1196-1216
ISSN 0010-437X
Publication year 2017
Published at Department of Mathematical Sciences, Algebra and Geometry
Pages 1196-1216
Language en
Keywords zeros of L-functions, Katz-Sarnak heuristics, quadratic Dirichlet L-functions, 1-level density, RANDOM-MATRIX THEORY, RATIOS CONJECTURE, ELLIPTIC-CURVES
Subject categories Mathematics


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 phi is contained in (-2, 2). We uncover a phase transition when the supremum sigma of the support of (phi) over cap reaches 1, both in the main term and in the lower order terms. A new lower order term appearing at sigma = 1 involves the quantity (phi) over cap (1), and is analogous to a lower order term which was isolated by Rudnick in the function field case.

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