High pseudomoments of the Riemann zeta function

TL;DR

This paper refines bounds for the pseudomoments of the Riemann zeta function, establishing precise asymptotics and exploring the behavior across different ranges of the parameter k.

Contribution

It improves the bounds for the constants in the asymptotic estimate of the pseudomoments and determines the first two terms of their asymptotic expansion.

Findings

Established sharper bounds for pseudomoments constants.

Determined the first two terms of the asymptotic expansion.

Analyzed the uniform ranges of k where the estimates hold.

Abstract

The pseudomoments of the Riemann zeta function, denoted $\mathcal{M}_k(N)$, are defined as the $2k$th integral moments of the $N$th partial sum of $\zeta(s)$ on the critical line. We improve the upper and lower bounds for the constants in the estimate $\mathcal{M}_k(N) \asymp_k (\log{N})^{k^2}$ as $N\to\infty$ for fixed $k\geq1$, thereby determining the two first terms of the asymptotic expansion. We also investigate uniform ranges of $k$ where this improved estimate holds and when $\mathcal{M}_k(N)$ may be lower bounded by the $2k$th power of the $L^\infty$ norm of the $N$th partial sum of $\zeta(s)$ on the critical line.