On the random Chowla conjecture

TL;DR

This paper proves that for certain polynomial sequences, the sum of a Steinhaus random multiplicative function converges in distribution to a complex Gaussian, confirming a conjecture and revealing Gaussian behavior for non-linear polynomial phases.

Contribution

It establishes the Gaussian limit law for sums of Steinhaus functions over polynomial sequences of degree at least two, confirming Najnudel's conjecture in a strong form.

Findings

Sum of Steinhaus functions over non-linear polynomials converges to complex Gaussian distribution.

Almost sure existence of large fluctuations matching the law of the iterated logarithm.

Contrasts with linear case where sums exhibit non-Gaussian behavior.

Abstract

We show that for a Steinhaus random multiplicative function $f:\mathbb{N}\to\mathbb{D}$ and any polynomial $P(x)\in\mathbb{Z}[x]$ of $\text{deg}\ P\ge 2$ which is not of the form $w(x+c)^{d}$ for some $w\in \mathbb{Z}$, $c\in \mathbb{Q}$, we have \[\frac{1}{\sqrt{x}}\sum_{n\le x} f(P(n)) \xrightarrow{d} \mathcal{CN}(0,1),\] where $\mathcal{CN}(0,1)$ is the standard complex Gaussian distribution with mean $0$ and variance $1.$ This confirms a conjecture of Najnudel in a strong form. We further show that there almost surely exist arbitrary large values of $x\ge 1,$ such that $$|\sum_{n\le x} f(P(n))| \gg_{\text{deg}\ P} \sqrt{x} (\log \log x)^{1/2},$$ for any polynomial $P(x)\in\mathbb{Z}[x]$ with $\text{deg}\ P\ge 2,$ which is not a product of linear factors (over $\mathbb{Q}$). 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 $\varepsilon>0$.