Random Chowla's Conjecture for Rademacher Multiplicative Functions

We study the distribution of partial sums of Rademacher random multiplicative functions $(f(n){)}_n$ evaluated at polynomial arguments. We show that for a polynomial $P\in \mathbb{Z}[x]$ that is a product of at least two distinct linear factors or an irreducible quadratic satisfying a natural condition, there exists a constant ${\kappa }_P>0$ such that $$\frac{1}{\sqrt{{\kappa }_{P}N}}\sum _{n\le N}f(P(n))\stackrel{d}{\to }\mathcal{N}(0,1),$$ as $N\to \mathrm{\infty }$, where convergence is in distribution to a standard (real) Gaussian. This confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu. We also study large fluctuations of ${\sum }_{n\le N}f(n^2+1)$ and show that there almost surely exist arbitrarily large values of $N$ such that $$\mid \sum _{n\le N}f(n^2+1)\mid \gg \sqrt{N\log \log N}\mathrm{.}$$ This matches the bound one expects from the law of iterated logarithm.