A unitary test of the Ratios Conjecture

The Ratios Conjecture of Conrey, Farmer and Zirnbauer (2008) [CFZ1], , (preprint) [CFZ2], 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 (Huynh and Miller (preprint) [HuyMil], , Miller (2009) [Mil5], , Miller and Montague (in press) [MilMo], ) and symplectic (Miller (2008) [Mil3], , Stopple (2009) [St]) 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.