Almost sure upper bound for random multiplicative functions

Abstract

Let $\epsilon >0$. Let $f$ be a Steinhaus or Rademacher random multiplicative function. We prove that we have almost surely, as $x\to +\infty$, $$\sum _{n⩽x}f(n)\ll \sqrt{x}({\log }_{2}x{)}^{\frac{3}{4}+\epsilon }.$$