Random Chowla's Conjecture for Rademacher Multiplicative Functions

Abstract

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 $\hat{I}{º}_P>0$ such that \[ \frac{1}{\sqrt{κ_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1), \] as $N\to \infty$, 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 \[ \Big|\sum_{n\leq N}f(n^2+1)\Big|\gg \sqrt{N \log\log N}. \] This matches the bound one expects from the law of iterated logarithm.

25 pages, accepted to Transactions of the AMS