Random polynomials, random matrices, and $L$-functions

TL;DR

This paper demonstrates how the Circular Orthogonal Ensemble of random matrices emerges from random polynomials, providing insight into the connection between random matrix theory and the zeros of the Riemann zeta-function.

Contribution

It reveals a natural origin of the Circular Orthogonal Ensemble in random polynomials, linking it to the statistical behavior of zeta-function zeros.

Findings

Circular Orthogonal Ensemble arises from random polynomials

Connection between random matrices and zeta-function zeros clarified

Provides a new perspective on random matrix statistics in number theory

Abstract

We show that the Circular Orthogonal Ensemble of random matrices arises naturally from a family of random polynomials. This sheds light on the appearance of random matrix statistics in the zeros of the Riemann zeta-function.