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.

A recent conjecture of Fyodorov-Hiary-Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is exp{log log T − 3 4 log log log T + O(1)}, for an interval at (large) height T . In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.2000 Mathematics Subject Classification. 60G70, 11M06.

We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We present an extended application to a Gaussian version of a random process studied by Halász. This leads to much improved lower bound results for the sum of a random multiplicative function. We further illustrate our methods by improving lower bounds for some classical constants from extreme value theory, the Pickands constants Hα, as α → 0.

We prove, assuming the Riemann Hypothesis, thatfor any fixed k ≥ 0 and all large T . This is sharp up to the value of the implicit constant.Our proof builds on well known work of Soundararajan, who showed, assuming the Riemann Hypothesis, thatT for any fixed k ≥ 0 and ǫ > 0. Whereas Soundararajan bounded log |ζ(1/2 + it)| by a single Dirichlet polynomial, and investigated how often it attains large values, we bound log |ζ(1/2+it)| by a sum of many Dirichlet polynomials and investigate the joint behaviour of all of them. We also work directly with moments throughout, rather than passing through estimates for large values.

Halász's Theorem gives an upper bound for the mean value of a multiplicative function f . The bound is sharp for general such f , and, in particular, it implies that a multiplicative function with |f (n)| ≤ 1 has either mean value 0, or is "close to" n it for some fixed t. The proofs in the current literature have certain features that are difficult to motivate and which are not particularly flexible. In this article we supply a different, more flexible, proof, which indicates how one might obtain asymptotics, and can be modified to short intervals and to arithmetic progressions. We use these results to obtain new, arguably simpler, proofs that there are always primes in short intervals (Hoheisel's Theorem), and that there are always primes near to the start of an arithmetic progression (Linnik's Theorem).