Moments of Random Multiplicative Functions, I: Low Moments, Better Than Squareroot Cancellation, and Critical Multiplicative Chaos

Abstract

We determine the order of magnitude of E| n≤x f (n)| 2q , where f (n) is a Steinhaus or Rademacher random multiplicative function, and 0 ≤ q ≤ 1. In the Steinhaus case, this is equivalent to determining the order of lim T →∞In particular, we find that E| n≤x f (n)| ≍ √ x/(log log x) 1/4 . This proves a conjecture of Helson that one should have better than squareroot cancellation in the first moment, and disproves counter-conjectures of various other authors. We deduce some consequences for the distribution and large deviations of n≤x f (n).The proofs develop a connection between E| n≤x f (n)| 2q and the q-th moment of a critical, approximately Gaussian, multiplicative chaos, and then establish the required estimates for that. We include some general introductory discussion about critical multiplicative chaos to help readers unfamiliar with that area.

Moments of random multiplicative functions, II: High moments

“…There are also probabilistic and analytic motivations for studying them, see Saksman and Seip's open problems paper [18], for example. The introduction to the previous paper [8] in this sequence contains a more extensive discussion of some of these connections.…”

“…Harper [8] showed that for Steinhaus or Rademacher random multiplicative f (n), for all large x we have…”

“…We can also write F (s) as an Euler product, namely F (s) = p≤x (1 − f (p) p s ) −1 in the Steinhaus case and F (s) = p≤x (1 + f (p) p s ) in the Rademacher case. In the author's treatment [8] of low moments, the first step was to show (roughly) that…”

“…Note the shift by q/ log x in the integral, which is analogous to the use of Rankin's trick in our elementary upper bound argument for q ≥ log log x. The basic strategy for proving something like (1.2) is the same as in [8], namely conditioning on the behaviour of f (n) on smaller primes; using fairly standard moment inequalities, like Khintchine's inequality, to show the conditional expectation behaves like a power of a mean square average; and using Parseval's identity to relate the mean square to an integral average of the Euler product. In [8] one could bound terms by using Hölder's inequality to pass to the second moment, whereas here we need suitable rough bounds for high moments.…”

Pseudomoments of the Riemann zeta function

“…Using this terminology, we may think of Z N as a sum of random multiplicative functions. A probabilistic approach to problems of computing integral moments, based on this viewpoint, can be found in recent work of Harper [11,13] and Harper, Nikeghbali and Radziwi l l [14]. In many situations, both approaches apply equally well, but sometimes one viewpoint is more illuminating and profitable than the other, and occasionally it is useful to combine them.…”