The $n$-level densities of low-lying zeros of quadratic Dirichlet $L$-functions
Volume 161 / 2013
Acta Arithmetica 161 (2013), 145-182
MSC: Primary 11M26, 15B52; Secondary 11M50.
DOI: 10.4064/aa161-2-3
Abstract
Previous work by Rubinstein and Gao computed the $n$-level densities for families of quadratic Dirichlet $L$-functions for test functions $\widehat{f}_1, \dots, \widehat{f}_n$ supported in $\sum_{i=1}^n |u_i| < 2$, and showed agreement with random matrix theory predictions in this range for $n \le 3$ but only in a restricted range for larger $n$. We extend these results and show agreement for $n \le 7$, 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.
