Moments of random multiplicative functions, II: High moments

Harper, Adam J

doi:10.2140/ant.2019.13.2277

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Moments of random multiplicative functions, II: High moments

Adam J Harper

Vol. 13 (2019), No. 10, 2277–2321

DOI: 10.2140/ant.2019.13.2277

Abstract

We determine the order of magnitude of $E {| \sum_{n \leq x} f (n) |}^{2 q}$ up to factors of size $e^{O (q^{2})}$ , where $f (n)$ is a Steinhaus or Rademacher random multiplicative function, for all real $1 \leq q \leq c log x ∕ log log x$ .

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

The proofs use various tools, including hypercontractive inequalities, to connect $E {| \sum_{n \leq x} f (n) |}^{2 q}$ 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.

Keywords

random multiplicative functions, random Euler products, moments, orthogonal behavior, unitary behavior, martingales

Mathematical Subject Classification 2010

Primary: 11N56

Secondary: 11K65, 11L40

Milestones

Received: 27 April 2018

Revised: 24 April 2019

Accepted: 5 July 2019

Published: 6 January 2020

Authors

Adam J Harper
University of Warwick Coventry United Kingdom

Vol. 13, No. 10, 2019