Optimal test functions for detecting central point vanishing over families of L-functions

Abstract

The Katz-Sarnak Density Conjecture states that on a large scale, central point vanishing of suitably indexed L-functions is well-modeled by eigenvalues of random matrices drawn from the classical compact groups. In particular, over suitably restricted eigenvalues, an average order vanishing can bounded using only a chosen test function ϕ.For each type of symmetry, we find the optimal ϕ, and we give very explicit descriptions for small support. We compute the optimal bound on average rank, an improvement on the bounds in the existing literature.In general, we show that, for a given support, the optimal test function lies in a finite-dimensional family. We prove this using a new method of solving certain integral equations on finite intervals. We differentiate under the integral sign and solve the resulting system of delay differential equations using a priori symmetries of the solution. This method may prove useful in a variety of applications.