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On the random Chowla conjecture

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Abstract

We show that for a Steinhaus random multiplicative function \(f:{\mathbb {N}}\rightarrow {\mathbb {D}}\) and any polynomial \(P(x)\in {\mathbb {Z}}[x]\) of \(\deg P\ge 2\) which is not of the form \(w(x+c)^{d}\) for some \(w\in {\mathbb {Z}}\), \(c\in {\mathbb {Q}}\), we have

$$\begin{aligned} \frac{1}{\sqrt{N}}\sum _{n\le N} f(P(n)) \xrightarrow {d} {{\mathcal {C}}}{{\mathcal {N}}}(0,1), \end{aligned}$$

where \({{\mathcal {C}}}{{\mathcal {N}}}(0,1)\) 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

$$\begin{aligned} \left| \sum _{n\le x} f(P(n))\right| \gg _{P} \sqrt{x} (\log \log x)^{1/2}, \end{aligned}$$

for any polynomial \(P(x)\in {\mathbb {Z}}[x]\) with \(\deg P\ge 2,\) 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 polynomial \(P(n)=n,\) 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 \(\varepsilon >0\).

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Notes

  1. In a recent preprint [WX22], the authors have made progress towards this question.

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Acknowledgements

The authors are very grateful to Adam Harper and Alexander Mangerel for the insightful comments on an earlier draft of this manuscript. We would like to warmly thank anonymous referees for a careful reading of the paper and for the numerous very helpful comments and corrections. O.K. and I.S. would like to thank Max Planck Institute for Mathematics (Bonn) for providing excellent working conditions and support. O.K. would also like to express his gratitude to Mittag-Leffler Institute for mathematical research for providing stimulating working environment and support. I.S. was supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-02-2022-872). M.W.X. is supported by the Cuthbert C. Hurd Graduate Fellowship in the Mathematical Sciences (Stanford). O.K. and I.S. are grateful for the FRG support of the Heilbronn Institute for Mathematical Research.

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Klurman, O., Shkredov, I.D. & Xu, M.W. On the random Chowla conjecture. Geom. Funct. Anal. 33, 749–777 (2023). https://doi.org/10.1007/s00039-023-00641-y

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