MOMENTS OF RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, AND CRITICAL MULTIPLICATIVE CHAOS

We determine the order of magnitude of Ej P n 6 x f.n /j2q, where f.n / is a Steinhaus or Rademacher random multiplicative function, and 0 6 q 6 1. In the Steinhaus case, this is equivalent to determining the order of limT !1 1 T R T 0 j P n 6x n it j2q dt. In particular, we find that Ej P n 6x f.n /j p 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 P n 6x f.n /. The proofs develop a connection between Ej P n 6 x f.n /j2q and the qth 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.