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

Abstract

Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of L-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of primitive Dirichlet L-functions matches up with corresponding statistic in the random unitary ensemble, in a range that includes the off-diagonal contribution. To estimate the n-level density of zeros of the Lfunctions, we use the asymptotic large sieve method developed by Conrey, Iwaniec and Soundararajan. For the random matrix side, a formula from Conrey and Snaith allows us to solve the matchup problem.