2009/09/30 by John Goes, Steven Jackson, Steven Glenn Jackson +6
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Cardinality (data modeling) #Combinatorics #Conjecture #Counterexample #Discrete mathematics #Geometry #Limits and Structures in Graph Theory #Mathematics #Preprint #Prime (order theory) #Prime power #Pure mathematics #Square (algebra) #Square root #Symplectic geometry #Unitary state #math.NT #msc:11M26 #msc:11M41 #msc:15B52
paper · pdf · doi:10.1016/j.jnt.2010.02.020
published as Journal of Number Theory 130 (2010), pp. 2238-2258 · Version 2: 24 pages, provided additional details, fixed some small mistakes and expanded the exposition in places
arxiv created 2010/01/22 · openalex publication_date 2010/06/14 · arxiv updated 2010/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.