Vol. 9, No. 1, 2015

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Surpassing the ratios conjecture in the 1-level density of Dirichlet $L$-functions

Daniel Fiorilli and Steven J. Miller

Vol. 9 (2015), No. 1, 13–52

We study the 1-level density of low-lying zeros of Dirichlet L-functions in the family of all characters modulo q, with Q2 < q Q. For test functions whose Fourier transform is supported in (3 2, 3 2), we calculate this quantity beyond the square root cancellation expansion arising from the L-function ratios conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the ratios conjecture, and proves that the exponent of the error term Q12+ϵ in the ratios conjecture is best possible. We also give more precise results when the support of the Fourier transform of the test function is restricted to the interval [1,1]. Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery’s, which implies that the ratios conjecture’s prediction holds for any finite support up to an error Q12+ϵ.

low-lying zeros, Dirichlet L-functions, ratios conjecture, primes in arithmetic progressions, random matrix theory
Mathematical Subject Classification 2010
Primary: 11M26, 11M50, 11N13
Secondary: 11N56, 15B52
Received: 27 September 2013
Revised: 2 April 2014
Accepted: 24 May 2014
Published: 18 February 2015
Daniel Fiorilli
Department of Mathematics
University of Michigan
530 Church Street
Ann Arbor, MI 48109
United States
Steven J. Miller
Mathematics and Statistics
Williams College
18 Hoxsey St
Bronfman Science Center
Williamstown, MA 01267
United States