Vol. 13, No. 10, 2019

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Moments of random multiplicative functions, II: High moments

Adam J Harper

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

We determine the order of magnitude of E| nxf(n)|2q up to factors of size eO(q2) , where f(n) is a Steinhaus or Rademacher random multiplicative function, for all real 1 q clogxloglogx.

In the Steinhaus case, we show that E| nxf(n)|2q = eO(q2) xq(logx(qlog(2q)))(q1)2 on this whole range. In the Rademacher case, we find a transition in the behavior of the moments when q (1 + 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 nxf(n).

The proofs use various tools, including hypercontractive inequalities, to connect E| nxf(n)|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 Lp maximal inequality for martingales.

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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