Orthogonal and symplectic n-level densities

TL;DR

This paper extends the application of the Ratios Conjectures to predict the n-level densities of zeros of L-functions with orthogonal or symplectic symmetry, building on methods used for the Riemann zeta function.

Contribution

It develops explicit conjectural formulas for the n-level densities of zeros of L-functions with orthogonal or symplectic symmetry, including lower order terms, and adapts methods for different test function supports.

Findings

Derived explicit conjectures for n-level densities of L-functions with orthogonal/symplectic symmetry.

Extended the method to include lower order terms in the density calculations.

Provided formulas adaptable to various test function supports.

Abstract

In this paper we apply to the zeros of families of $L$-functions with orthogonal or symplectic symmetry the method that Conrey and Snaith used to calculate the $n$-correlation of the zeros of the Riemann zeta function. This method uses the Ratios Conjectures for averages of ratios of zeta or $L$-functions. Katz and Sarnak conjecture that the zero statistics of families of $L$-functions have an underlying symmetry relating to one of the classical compact groups $U(N)$, $O(N)$ and $USp(2N)$. Here we complete the work already done with $U(N)$ to show how new methods for calculating the $n$-level densities of eigenangles of random orthogonal or symplectic matrices can be used to create explicit conjectures for the $n$-level densities of zeros of $L$-functions with orthogonal or symplectic symmetry, including all the lower order terms. We show how the method used here results in formulae that are easily modified when the test function used has a restricted range of support, and this will facilitate comparison with rigorous number theoretic $n$-level density results.