# Pseudomoments of the Riemann zeta function

## TL;DR

This paper investigates the growth of pseudomoments of the Riemann zeta function, providing new inequalities and bounds, especially for cases where the moment order is less than 1/2, revealing unexpected lower bounds.

## Contribution

It introduces new Hardy--Littlewood inequalities and improves asymptotic estimates for pseudomoments of the zeta function, especially for small moment orders.

## Key findings

- Pseudomoments grow like (log N)^{k^2} for k > 1/2.
- New inequalities are derived to estimate growth rates.
- Unexpected lower bounds are found for pseudomoments when k<1/e.

## Abstract

The $2$kth pseudomoments of the Riemann zeta function $\zeta(s)$ are, following Conrey and Gamburd, the $2k$th integral moments of the partial sums of $\zeta(s)$ on the critical line. For fixed $k>1/2$, these moments are known to grow like $(\log N)^{k^2}$, where $N$ is the length of the partial sum, but the true order of magnitude remains unknown when $k\le 1/2$. We deduce new Hardy--Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when $k\to\infty$. In the case $k<1/2$, we consider pseudomoments of $\zeta^{\alpha}(s)$ for $\alpha>1$ and the question of whether the lower bound $(\log N)^{k^2\alpha^2}$ known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali, and Radziwi{\l\l} and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by $\log N$ to a power larger than $k^2\alpha^2$ when $k<1/e$ and $N$ is sufficiently large.