On the random Chowla conjecture
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 wellknown case of linear phase $P(n)=n,$ where the partial sums are known to behave in a nonGaussian fashion and the corresponding sharp fluctuations are speculated to be $O(\sqrt{x}(\log \log x)^{\frac{1}{4}+\varepsilon})$ for any $\varepsilon>0$.
 Publication:

arXiv eprints
 Pub Date:
 February 2022
 DOI:
 10.48550/arXiv.2202.08767
 arXiv:
 arXiv:2202.08767
 Bibcode:
 2022arXiv220208767K
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Probability
 EPrint:
 Minor changes to the Introduction. Added remark about possibility of extending the proofs to sparse sets. Added references to the work of CassaigneFerencziMauduitRivatSarkozy and BorweinChoiGanguly on the sign changes of $\lambda (P(n)).$ All other sections remain unchanged