A unitary test of the Ratios Conjecture

TL;DR

This paper provides a rigorous test of the Ratios Conjecture for the family of Dirichlet L-functions with prime conductor, confirming its predictions with high precision for certain test function supports.

Contribution

It verifies the Ratios Conjecture's predictions for the unitary family of Dirichlet L-functions, extending previous results to broader support ranges and achieving power savings.

Findings

Confirmed the conjecture's predictions for support in (-1,1)

Achieved agreement up to a power savings for support in (-2,2)

Extended verification to the family of Dirichlet L-functions with prime conductor

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.