The n-level densities of low-lying zeros of quadratic Dirichlet L-functions

TL;DR

This paper extends the analysis of low-lying zeros of quadratic Dirichlet L-functions to higher n-level densities, showing agreement with random matrix theory predictions up to n<8, using a new combinatorial approach.

Contribution

It advances previous results by increasing the range of n for which the density matches predictions and introduces a novel combinatorial method for the comparison.

Findings

Agreement with random matrix theory for n<8

Reduction of higher n to a Fourier transform identity

Introduction of a new combinatorial perspective

Abstract

Previous work by Rubinstein and Gao computed the n-level densities for families of quadratic Dirichlet L-functions for test functions where the sum of the supports of the Fourier transforms is at most 2, and showed agreement with random matrix theory predictions in this range for n < 4 but only in a restricted range for larger n. We extend these results and show agreement for n < 8, and reduce higher n to a Fourier transform identity. The proof involves adopting a new combinatorial perspective to convert all terms to a canonical form, which facilitates the comparison of the two sides.