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 $q> 0$.

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

The goal of this work is to prove an analogue of a recent result of Harper on almost sure lower bounds of random multiplicative functions, in a setting that can be thought of as a simplified function field analogue. It answers a question raised in the work of Soundararajan and Zaman, who proved moment bounds for the same quantity in analogy to those of Harper in the random multiplicative setting. Having a simpler quantity allows us to make the proof close to self‐contained, and perhaps somewhat more accessible.

The goal of this work is to prove an analogue of a recent result of Harper on almost sure lower bounds of random multiplicative functions, in a setting that can be thought of as a simplified function field analogue. It answers a question raised in work of Soundararajan and Zaman, who proved moment bounds for the same quantity in analogy to those of Harper in the random multiplicative setting. Having a simpler quantity allows us to make the proof close to self-contained, and perhaps somewhat more accessible.

The 2q-th pseudomoment Ψ2q,α(x) of the α-th power of the Riemann zeta function is defined to be the 2q-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 ≤ 1 2 and α ≥ 1. Combined with results of Bondarenko, Heap and Seip, these bounds determine the size of all pseudomoments with q > 0 and α ≥ 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 Ψ2q,1(x) for all q > 0.