Symplectic n-level densities with restricted support

Abstract

In this paper we demonstrate that the alternative form, derived by us in an earlier paper, of the n-level densities for eigenvalues of matrices from the classical compact group U Sp(2N ) is far better suited for comparison with derivations of the n-level densities of zeros in the family of Dirichlet L-functions associated with real quadratic characters than the traditional determinantal random matrix formula. Previous authors have found ingenious proofs that the leading order term of the n-level density of the zeros agrees with the determinantal random matrix result under certain conditions, but here we show that comparison is more straightforward if the more suitable form of the random matrix result is used. For the support of the test function in [1, −1] and in [−2, 2] we compare with existing number theoretical results. For support in [−3, 3] no rigorous number theoretical result is known for the n-level densities, but we derive the densities here using random matrix theory in the hope that this may make the path to a rigorous number theoretical result clearer.

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

“…This was then resolved by Entin, Roditty-Gershon and Rudnick through zeta functions over function fields [8]. Recently, Mason and Snaith [15] presented an alternative proof of this result using a new formula for n-level densities of the random symplectic ensemble, analogous to the work of Conrey and Snaith in [5] and [6].…”

Lower-Order Terms of the One-Level Density of a Family of Quadratic Hecke -Functions

“…This approach was applied by Conrey and Snaith in [5] to study the one-level density function for zeros of quadratic Dirichlet L -functions. The general n -level density of the same family was examined by Mason and Snaith in [27], and further enabled them to show in [26] that the result agrees with the density conjecture when the Fourier transforms of test functions are supported in .…”