Random Chowla's Conjecture for Rademacher Multiplicative Functions

TL;DR

This paper proves that partial sums of Rademacher multiplicative functions evaluated at certain polynomial arguments converge in distribution to a Gaussian, confirming a conjecture and analyzing large fluctuations for specific polynomial sequences.

Contribution

It establishes Gaussian distribution results for sums of Rademacher functions at polynomial arguments and investigates large fluctuations for quadratic polynomials, confirming conjectures and addressing open questions.

Findings

Distribution of sums converges to Gaussian for specified polynomials.

Existence of arbitrarily large fluctuations matching the law of iterated logarithm.

Confirms a conjecture of Najnudel and addresses a question of Klurman-Shkredov-Xu.

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 $\kappa_P>0$ such that \[ \frac{1}{\sqrt{\kappa_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1), \] as $N\rightarrow\infty$, 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 $\sum_{n\leq N}f(n^2+1)$ 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.