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 ε>0 small. We prove that, as x→+∞, we almost surely have | ∑ n≤xP(n)>xf(n)|≤ x(loglogx)1∕4+ε, 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.