Moments of polynomials with random multiplicative coefficients

Abstract

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$P_N(\theta )=\frac{1}{\sqrt{N}}\sum _{n\le N}X(n)e(n\theta ),$$ and show that the $2k$-th moments on the unit circle $${\int }_0^1|P_N(\theta ){|}^{2k}d\theta$$ tend to Gaussian moments in the sense of mean-square convergence, uniformly for $k\ll (\log N/\log \log N{)}^{1/3}$, but that in contrast to the case of i.i.d. 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(i) give a proof of an almost sure Salem-Zygmund type central limit theorem for $P_N(\theta )$, previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely $$(\log N{)}^{1/6-\epsilon }\ll \underset{\theta }{\max }|P_N(\theta )|\ll \exp ((\log N{)}^{1/2+\epsilon }),$$ for all $\epsilon >0$.

Minor changes in Theorem 1.4