Moments of random multiplicative functions, II: High moments

Abstract

We determine the order of magnitude of $\mathbb{E}|{\sum }_{n\le x}f(n){|}^{2q}$ up to factors of size $e^{O(q^2)}$, where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1\le q\le \frac{c\log x}{\log \log x}$. In the Steinhaus case, we show that $\mathbb{E}|{\sum }_{n\le x}f(n){|}^{2q}=e^{O(q^2)}{x}^q(\frac{\log x}{q\log (2q)}{)}^{(q-1{)}^2}$ 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 ${\sum }_{n\le x}f(n)$. The proofs use various tools, including hypercontractive inequalities, to connect $\mathbb{E}|{\sum }_{n\le x}f(n){|}^{2q}$ 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.

45 pages

References in corpus (2)

• Shifted moments of L functions and moments of theta functions
• Pseudomoments of the Riemann zeta function

Cited by in corpus (9)

• Moments of polynomials with random multiplicative coefficients
• Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function
• Partial sums of typical multiplicative functions over short moving intervals
• Paucity phenomena for polynomial products
• Almost sure bounds for a weighted Steinhaus random multiplicative function
• Magic squares, the symmetric group and Möbius randomness
• Bounds for exponential sums with random multiplicative coefficients
• Random Chowla's Conjecture for Rademacher Multiplicative Functions
• Moments of random multiplicative functions over function fields