On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices

TL;DR

This paper demonstrates that nearly all zeros of finite linear combinations of independent characteristic polynomials of random unitary matrices are on the unit circle, supporting conjectured links with $L$-functions.

Contribution

It extends previous results by showing zeros of these linear combinations predominantly lie on the unit circle, analogous to $L$-function zero distributions.

Findings

Zeros mostly on the unit circle

Supports conjectured link between random matrices and $L$-functions

Provides evidence for value distribution similarities

Abstract

We show that almost all the zeros of any finite linear combination of independent characteristic polynomials of random unitary matrices lie on the unit circle. This result is the random matrix analogue of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of $L$-functions, thus providing further evidence for the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of $L$-functions on the critical line.