In support of $n$-correlation

TL;DR

This paper investigates $n$-correlation of eigenvalues and zeros of $L$-functions with limited Fourier support, providing new formulas that clarify which terms remain under support restrictions, extending previous work by Rudnick and Sarnak.

Contribution

It introduces a new expression for $n$-correlation that applies to arbitrary support, simplifying the analysis of surviving terms under Fourier transform limitations.

Findings

Derived a new $n$-correlation formula applicable to arbitrary support

Clarified the terms that survive support restrictions in $n$-correlation

Extended the connection between random matrix theory and number theory results

Abstract

In this paper we examine $n$-correlation for either the eigenvalues of a unitary group of random matrices or for the zeros of a unitary family of $L$-functions in the important situation when the correlations are detected via test functions whose Fourier transforms have limited support. This problem first came to light in the work of Rudnick and Sarnak in their study of the $n$-correlation of zeros of a fairly general automorphic $L$-function. They solved the simplest instance of this problem when the total support was most severely limited, but had to work extremely hard to show their result matched random matrix theory in the appropriate limit. This is because they were comparing their result to the familiar determinantal expressions for $n$-correlation that arise naturally in random matrix theory. In this paper we deal with arbitrary support and show that there is another expression for the $n$-correlation of eigenvalues that translates easily into the number theory case and allows for immediate identification of which terms survive the restrictions placed on the support of the test function.