Low Pseudomoments of the Riemann Zeta Function and Its Powers
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 et al., 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 $q> 0$.