A sharp almost sure upper bound for partial sums of random multiplicative functions

Abstract

We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\epsilon >0$, almost surely $|{\sum }_{n\le x}f(n)|{\ll }_{\epsilon ,f}\sqrt{x}(\log \log x{)}^{1/4+\epsilon }$. Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.

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