A unitary test of the Ratios Conjecture

Details A unitary test of the Ratios Conjecture A unitary test of the Ratios Conjecture

2009-09-27

arxiv.org/abs/0909.4916v3

Journal of Number Theory 130 (2010), pp. 2238-2258

Mathematics

Number Theory

Version 2: 24 pages, provided additional details, fixed some small mistakes and expanded the exposition in places

Scientific paper

10.1016/j.jnt.2010.02.020

The Ratios Conjecture of Conrey, Farmer and Zirnbauer predicts the answers to numerous questions in number theory, ranging from n-level densities and correlations to mollifiers to moments and vanishing at the central point. The conjecture gives a recipe to generate these answers, which are believed to be correct up to square-root cancelation. These predictions have been verified, for suitably restricted test functions, for the 1-level density of orthogonal and symplectic families of L-functions. In this paper we verify the conjecture's predictions for the unitary family of all Dirichlet $L$-functions with prime conductor; we show square-root agreement between prediction and number theory if the support of the Fourier transform of the test function is in (-1,1), and for support up to (-2,2) we show agreement up to a power savings in the family's cardinality.

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