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

Abstract

ABSTRACT. Previous work by Rubinstein [Rub] and Gao [Gao] computed the n-level densities for families of quadratic Dirichlet L-functions for test functions φ1, . . . , φn supported in ∑ n i=1 |ui| < 2, and showed agreement with random matrix theory predictions in this range for n ≤ 3 but only in a restricted range for larger n. We extend these results and show agreement for n ≤ 7, 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.