High pseudomoments of the Riemann zeta function

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 $k\ge 1$, 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.

This paper has been accepted for publication in Journal of Number Theory