Uniform asymptotics for the full moment conjecture of the Riemann zeta function

TL;DR

This paper derives uniform asymptotics for the initial coefficients of the moment polynomial of the Riemann zeta function, supporting the full moment conjecture with numerical evidence and applications to bounding zeta's maximum size.

Contribution

It provides the first uniform asymptotic formulas for the initial coefficients of the zeta moment polynomial, advancing understanding of the conjectured formulas.

Findings

Asymptotic formulas for the first $k$ coefficients are established.

Numerical data supports the accuracy of the asymptotic formulas.

Application to bounds on the maximum size of the zeta function is demonstrated.

Abstract

Conrey, Farmer, Keating, Rubinstein, and Snaith, recently conjectured formulas for the full asymptotics of the moments of $L$-functions. In the case of the Riemann zeta function, their conjecture states that the $2k$-th absolute moment of zeta on the critical line is asymptotically given by a certain $2k$-fold residue integral. This residue integral can be expressed as a polynomial of degree $k^2$, whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first $k$ coefficients of the moment polynomial are derived. Numerical data to support our asymptotic formula are presented. An application to bounding the maximal size of the zeta function is considered.