Moments of random multiplicative functions, II: High moments

We determine the order of magnitude of ‫ޅ‬ n≤x f (n)2q up to factors of size e O(q 2 ) , where f (n) is a Steinhaus or Rademacher random multiplicative function, for all real 1 ≤ q ≤ c log x/ log log x.In the Steinhaus case, we show that))) (q-1) 2 on this whole range.In the Rademacher case, we find a transition in the behavior of the moments when q ≈ (1 + √ 5)/2, where the size starts to be dominated by "orthogonal" rather than "unitary" behavior.We also deduce some consequences for the large deviations of n≤x f (n).The proofs use various tools, including hypercontractive inequalities, to connect ‫ޅ‬ n≤x f (n)2q with the q-th moment of an Euler product integral.When q is large, it is then fairly easy to analyze this integral.When q is close to 1 the analysis seems to require subtler arguments, including Doob's L p maximal inequality for martingales.