Abstract
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.
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