$n$-level density of the low-lying zeros of quadratic Dirichlet $L$-functions

TL;DR

This paper extends the computation of the $n$-level density of low-lying zeros of quadratic Dirichlet $L$-functions, under the Generalized Riemann Hypothesis, to a broader support region, confirming the Density Conjecture in this new setting.

Contribution

It improves Rubinstein's previous results by extending the support of the Fourier transform to the region where the sum of variables is less than 2.

Findings

Confirmed the $n$-level density matches the Density Conjecture for larger support

Extended the range of test functions for which the density can be computed

Provided evidence supporting the Katz-Sarnak conjecture in this context

Abstract

The Density Conjecture of Katz and Sarnak associates a classical compact group to each reasonable family of $L$-functions. Under the assumption of the Generalized Riemann Hypothesis, Rubinstein computed the $n$-level density of low-lying zeros for the family of quadratic Dirichlet $L$-functions in the case that the Fourier transform $\hat{f}(u)$ of any test function $f$ is supported in the region $\sum^n_{j=1}u_j < 1$ and showed that the result agrees with the Density Conjecture. In this paper, we improve Rubinstein's result on computing the $n$-level density for the Fourier transform $\hat{f}(u)$ being supported in the region $\sum^n_{j=1}u_j < 2$.