The -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.