n-Level Density of the Low-lying Zeros of Quadratic Dirichlet L-Functions

The Density Conjecture of Katz and Sarnak associates a classical compact group to each reasonable family of L-functions. Under the assumption of the GRH, Rubinstein computed the n-level density of low-lying zeros for the family of quadratic Dirichlet L-functions in the case that the Fourier transform $\hat {f}(u)$ of any test function f is supported in the region $\Sigma^n_{j=1}|u_j|\lt 1$ and showed that the result agrees with the Density Conjecture. In this paper, we improve Rubinstein’s result on computing the n-level density for the Fourier transform $\hat {f}(u)$ being supported in the region $\Sigma^n_{j=1}|u_j|\lt 2$.