Moments of polynomials with random multiplicative coefficients

For X(n) a Rademacher or Steinhaus random multi- plicative function, we consider the random polynomials P-N(theta) = 1/root N Sigma(n <= N) X(n)e(n theta), and show that the 2kth moments on the unit circle integral(1)(0) vertical bar P-N(theta)vertical bar(2k) d theta tend to Gaussian moments in the sense of mean-square convergence, uniformly for k << (log N/ log log N)(1/3), but that in contrast to the case of independent and identically distributed coefficients, this behavior does not persist for k much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for P-N(theta), previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely (log N)(1/6-epsilon) << max(theta) vertical bar P-N(theta)vertical bar << exp((log N)(1/2+epsilon)), for all epsilon > 0.