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.

In this paper, we determine the order of magnitude of the 2 q-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 Gerspach. Our results also apply to more general Euler products satisfying certain conditions.