#### Vol. 13, No. 10, 2019

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Moments of random multiplicative functions, II: High moments

Vol. 13 (2019), No. 10, 2277–2321
##### Abstract

We determine the order of magnitude of $\mathbb{E}{|{\sum }_{n\le x}f\left(n\right)|}^{2q}$ 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 $\mathbb{E}{|{\sum }_{n\le x}f\left(n\right)|}^{2q}={e}^{O\left({q}^{2}\right)}{x}^{q}{\left(logx∕\left(qlog\left(2q\right)\right)\right)}^{{\left(q-1\right)}^{2}}$ 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 $\mathbb{E}{|{\sum }_{n\le x}f\left(n\right)|}^{2q}$ 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.

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