  For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we
consider the random polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq
N} X(n) e(n\theta), $$ and show that the $2k$-th moments on the unit circle $$
\int_0^1 \big| P_N(\theta) \big|^{2k}\, d\theta $$ 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$P_N(\theta)$, previously obtained in unpublished work of Harper by different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 $$ (\log N)^{1/6 -
\varepsilon} \ll \max_\theta |P_N(\theta)| \ll \exp((\log
N)^{1/2+\varepsilon}), $$ for all $\varepsilon > 0$.
