Moments of random multiplicative functions, II: High moments

We determine the order of magnitude of E vertical bar Sigma(n <= x) f(n)vertical bar(2q) up to factors of size e(O(q2)), 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 E vertical bar Sigma(n <= x) f(n)vertical bar(2q) = e(O(q2)) x(q)(log x/q log(2q)))((q-1)2) on this whole range. In the Rademacher case, we find a transition in the behavior of the moments when q approximate to (1 + root 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 Sigma(n <= x) f(n). The proofs use various tools, including hypercontractive inequalities, to connect E vertical bar Sigma(n <= x) f(n)vertical bar(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.