Low pseudomoments of Euler products

TL;DR

This paper determines the order of magnitude of certain pseudomoments of powers of the Riemann zeta function, extending previous results and applying to general Euler products under specific conditions.

Contribution

It provides the first precise asymptotic estimates for the 2q-th pseudomoments of zeta powers for 0<q≤1/2 and 0<α<1, generalizing prior work.

Findings

Established the order of magnitude of the 2q-th pseudomoments for specified parameters.

Extended results to a broader class of Euler products.

Completed the analysis initiated by previous researchers.

Abstract

In this paper, we determine the order of magnitude of the $2q$-th pseudomoment of powers of the Riemann zeta function $\zeta(s)^{\alpha}$ for $0<q\le 1/2$ and $0< \alpha<1$, completing the results of Bondarenko, Heap and Seip, and of Gerspach. Our results also apply to more general Euler products satisfying certain conditions.