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 [Formula: see text]-level densities for eigenvalues of matrices from the classical compact group [Formula: see text] is far better suited for comparison with derivations of the [Formula: see text]-level densities of zeros in the family of Dirichlet [Formula: see text]-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 [Formula: see text]-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 [Formula: see text] and in [Formula: see text] we compare with existing number theoretical results. For support in [Formula: see text] no rigorous number theoretical result is known for the [Formula: see text]-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.