A unitary test of the Ratios Conjecture

Abstract

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.

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

References in corpus (14)

• 1- and 2-Level Densities for Rational Families of Elliptic Curves: Evidence for the Underlying Group Symmetries
• Low-lying zeros of families of elliptic curves
• A Symplectic Test of the L-Functions Ratios Conjecture
• Nuclei, Primes and the Random Matrix Connection
• The effect of convolving families of L-functions on the underlying group symmetries
• Low lying zeros of L-functions with orthogonal symmetry
• $n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions
• The Low Lying Zeros of a GL(4) and a GL(6) family of L-functions
• An Orthogonal Test of the L-Functions Ratios Conjecture
• Lower-Order Terms of the 1-Level Density of Families of Elliptic Curves
• Variation in the number of points on elliptic curves and applications to excess rank
• An Orthogonal Test of the $L$-functions Ratios Conjecture, II
• A Hybrid Euler-Hadamard product formula for the Riemann zeta function
• Lower order terms in the 1-level density for families of holomorphic cuspidal newforms

Cited by in corpus (5)

• Nuclei, Primes and the Random Matrix Connection
• Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions
• An Orthogonal Test of the $L$-functions Ratios Conjecture, II
• Determining optimal test functions for $2$-level densities
• Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters