Moments of random multiplicative functions, II: High moments

Abstract

We determine the order of magnitude of $\mathbb{E}{\left|{\sum }_{n\le x}f\left(n\right)\right|}^{2q}$ up to factors of size $e^{O\left(q^2\right)}$, where $f\left(n\right)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1\le q\le clogx∕loglogx$.

In the Steinhaus case, we show that $\mathbb{E}{\left|{\sum }_{n\le x}f\left(n\right)\right|}^{2q}=e^{O\left(q^2\right)}{x}^q{\left(logx∕\left(qlog\left(2q\right)\right)\right)}^{{\left(q-1\right)}^2}$ on this whole range. In the Rademacher case, we find a transition in the behavior of the moments when $q\approx \left(1+\sqrt{5}\right)∕2$, where the size starts to be dominated by “orthogonal” rather than “unitary” behavior. We also deduce some consequences for the large deviations of ${\sum }_{n\le x}f\left(n\right)$.

The proofs use various tools, including hypercontractive inequalities, to connect $\mathbb{E}{\left|{\sum }_{n\le x}f\left(n\right)\right|}^{2q}$ with the $q$-th moment of an Euler product integral. When $q$ is large, it is then fairly easy to analyze this integral. When $q$ is close to 1 the analysis seems to require subtler arguments, including Doob’s $L^p$ maximal inequality for martingales.