Low pseudomoments of the Riemann zeta function and its powers

TL;DR

This paper establishes bounds for the pseudomoments of the Riemann zeta function's powers, revealing different growth behaviors across ranges and determining their order of magnitude for all positive q.

Contribution

It provides new bounds and growth rate classifications for the pseudomoments of zeta function powers, extending previous results with probabilistic methods.

Findings

Bounds for pseudomoments when q ≤ 1/2 and α ≥ 1

Identification of three growth regimes for pseudomoments

Order of magnitude of pseudomoments for all q > 0

Abstract

The $2 q$-th pseudomoment $\Psi_{2q,\alpha}(x)$ of the $\alpha$-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $\zeta^\alpha$ on the critical line. Using probabilistic methods of Harper, we prove upper and lower bounds for these pseudomoments when $q \le \frac{1}{2}$ and $\alpha \ge 1$. Combined with results of Bondarenko, Heap and Seip, these bounds determine the size of all pseudomoments with $q > 0$ and $\alpha \ge 1$ up to powers of $\log \log x$, where $x$ is the length of the partial sum, and it turns out that there are three different ranges with different growth behaviours. In particular, the results give the order of magnitude of $\Psi_{2 q, 1}(x)$ for all $q > 0$.