Uniform asymptotics of the coefficients of unitary moment polynomials

TL;DR

This paper derives uniform asymptotics for the coefficients of unitary moment polynomials, identifying maximal coefficients and exploring connections to the Riemann zeta function, supported by numerical data.

Contribution

It provides explicit uniform asymptotic formulas for polynomial coefficients related to random matrix theory, advancing understanding of their behavior and maxima.

Findings

Explicit asymptotic formulas for coefficients

Identification of maximal coefficients

Numerical data supporting asymptotic results

Abstract

Keating and Snaith showed that the $2k^{th}$ absolute moment of the characteristic polynomial of a random unitary matrix evaluated on the unit circle is given by a polynomial of degree $k^2$. In this article, uniform asymptotics for the coefficients of that polynomial are derived, and a maximal coefficient is located. Some of the asymptotics are given in explicit form. Numerical data to support these calculations are presented. Some apparent connections between random matrix theory and the Riemann zeta function are discussed.