On the random Chowla conjecture

We show that for a Steinhaus random multiplicative function $f$ and any polynomial $P(x)\in Z[x], P(x) \neq w(x+c)^{d}, w\in Z, c\in Q,$ we have that
$$\frac{1}{\sqrt{x}} \sum_{n\le x} f(P(n))$$
converges in distribution to $N(0,1).$ We further show that there almost surely exist arbitrary large values of x such that
$$|\sum_{n\le x} f(P(n))| \gg_{\deg P} \sqrt{x} (\log \log x)^{1/2},$$
for any polynomial $P$ which is not a product of linear factors (over $Q$). This matches the bound predicted by the law of the iterated logarithm. Both of these results are in contrast with the well-known case of linear phase $P(n)=n,$ where the partial sums are known to behave in a non-Gaussian fashion and the corresponding sharp fluctuations are speculated to be $O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon})$ for any $\varepsilon>0.$
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000