Symplectic $n$-level densities with restricted support

TL;DR

This paper introduces a new form of the $n$-level density for $USp(2N)$ matrices that simplifies comparison with number theory results, especially for restricted test function support, and extends predictions to support ranges lacking rigorous proofs.

Contribution

We propose an alternative, more suitable form of the $n$-level density for symplectic matrices, facilitating comparison with number theoretical results and extending predictions to larger support ranges.

Findings

New form of $n$-level density simplifies comparisons

Derived densities for support in $[-3,3]$ where no rigorous results exist

Demonstrated better alignment with number theory for 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 $USp(2N)$ 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.