Abstract
We determine the order of magnitude of $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ , where $f(n)$ is a Steinhaus or Rademacher random multiplicative function, and $0\leqslant q\leqslant 1$ . In the Steinhaus case, this is equivalent to determining the order of $\lim _{T\rightarrow \infty }\frac{1}{T}\int _{0}^{T}|\sum _{n\leqslant x}n^{-it}|^{2q}\,dt$ . In particular, we find that $\mathbb{E}|\sum _{n\leqslant x}f(n)|\asymp \sqrt{x}/(\log \log x)^{1/4}$ . 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 $\sum _{n\leqslant x}f(n)$ . The proofs develop a connection between $\mathbb{E}|\sum _{n\leqslant x}f(n)|^{2q}$ 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.
Highlights
We show that it suffices to prove a comparable statement for quantities like E| n x,P(n)>√x f (n)|2q, where P(n) denotes the largest prime factor of n. (For the lower bound, this is literally true since if n x,P(n)>√x f (n) is large, with positive conditional probability, the complete sum will be large
The model for log ζ (1/2 + it) is very close to log |F(1/2 + it)| in the Steinhaus case, so it is possible that combining (1), Saksman and Webb’s approximation and the results of Duplantier, Rhodes, Sheffield and Vargas [6] about moments of the total measure of critical Gaussian chaos, one could get another proof of Theorem 1 for q bounded away from 1
|h( j)| 10 log j, and with Il(s) denoting the increments of the Euler product corresponding to a Steinhaus random multiplicative function, we have j
Summary
The model for log ζ (1/2 + it) is very close to log |F(1/2 + it)| in the Steinhaus case, so it is possible that combining (the rigorous version of) (1), Saksman and Webb’s approximation and the results of Duplantier, Rhodes, Sheffield and Vargas [6] about moments of the total measure of critical Gaussian chaos (which stem from Kahane’s convexity inequality and results of Hu and Shi [17] for branching random walk), one could get another proof of Theorem 1 for q bounded away from 1. The books of Gut [8] and of Montgomery and Vaughan [20] may be consulted as excellent general references for probabilistic and number theoretic background
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