MOMENTS of RANDOM MULTIPLICATIVE FUNCTIONS, I: LOW MOMENTS, BETTER THAN SQUAREROOT CANCELLATION, and CRITICAL MULTIPLICATIVE CHAOS

Abstract

We determine the order of magnitude of, where is a Steinhaus or Rademacher random multiplicative function, and. In the Steinhaus case, this is equivalent to determining the order of. In particular, we find that. 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. The proofs develop a connection between and the 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.