Low pseudomoments of the Riemann zeta function and its powers

Abstract

The $2 q$-th pseudomoment $Ψ_{2q,α}(x)$ of the $α$-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $ζ^α$ 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 $α\ge 1$. Combined with results of Bondarenko, Heap and Seip, these bounds determine the size of all pseudomoments with $q > 0$ and $α\ge 1$ 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 $Ψ_{2 q, 1}(x)$ for all $q > 0$.