Superposition of zeros of distinct L-functions

Abstract

In this paper we first prove a weighted prime number theorem of an ''o¤-diagonal'' type for Rankin-Selberg L-functions of automorphic representations of GLm and GLm0 over Q. Then for ma 1, or under the Selberg orthonormality conjecture for mV 2, we prove that non- trivial zeros of distinct primitive automorphic L-functions for GLm over Q are uncorrelated, for certain test functions whose Fourier transforms have restricted support. For the same test functions, we also prove that the n-level correlation of non-trivial zeros of a product of such L-functions follows the distribution of the superposition of GUE models for individual L-functions and GUEs of lower ranks.