Moments of random multiplicative functions, II: High moments

TL;DR

This paper determines the order of magnitude of high moments of random multiplicative functions, revealing different behaviors for Steinhaus and Rademacher cases and implications for large deviations.

Contribution

It provides a precise asymptotic analysis of high moments of random multiplicative functions, including a phase transition in the Rademacher case, using advanced probabilistic tools.

Findings

Exact order of magnitude for moments of Steinhaus functions.

Identification of a phase transition in Rademacher moments at q ≈ (1+√5)/2.

Insights into large deviation behavior of sums of random multiplicative functions.

Abstract

We determine the order of magnitude of $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq \frac{c\log x}{\log\log x}$. In the Steinhaus case, we show that $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q} = e^{O(q^2)} x^q (\frac{\log x}{q\log(2q)})^{(q-1)^2}$ on this whole range. In the Rademacher case, we find a transition in the behaviour of the moments when $q \approx (1+\sqrt{5})/2$, where the size starts to be dominated by "orthogonal" rather than "unitary" behaviour. We also deduce some consequences for the large deviations of $\sum_{n \leq x} f(n)$. The proofs use various tools, including hypercontractive inequalities, to connect $\mathbb{E}|\sum_{n \leq x} f(n)|^{2q}$ with the $q$-th moment of an Euler product integral. When $q$ is large, it is then fairly easy to analyse this integral. When $q$ is close to 1 the analysis seems to require subtler arguments, including Doob's $L^p$ maximal inequality for martingales.