An almost sure upper bound for random multiplicative functions on integers with a large prime factor

Abstract

Let f be a Rademacher or a Steinhaus random multiplicative function. Let $\mathit{\epsilon }>0$ small. We prove that, as $x\to +\mathrm{\infty }$, we almost surely have

$$|\sum _{\begin{array}{c}n\le x\\ P(n)>\sqrt{x}\end{array}}f(n)|\le \sqrt{x}{(loglogx)}^{1∕4+\mathit{\epsilon }},$$

where $P(n)$ stands for the largest prime factor of n. This gives an indication of the almost sure size of the largest fluctuations of f.