Random Chowla's Conjecture for Rademacher Multiplicative Functions
Jake Chinis, Besfort Shala
TL;DR
This work proves a real Gaussian limit for sums of a Rademacher random multiplicative function evaluated at polynomial arguments $P(n)$ when $P$ is either a product of at least two distinct linear factors or an admissible irreducible quadratic, establishing a Rademacher analogue of a random Chowla-type CLT. The main novelty lies in a counting proposition for squarefree polynomial values, proved via a bootstrapping strategy that leverages Bombieri–Pila bounds and detailed divisor-congruence analyses. The CLT is obtained by decomposing the partial sum into a martingale difference array and verifying McLeish’s conditions through precise second- and fourth-moment estimates. In addition, the paper proves large-fluctuation results for $\sum_{n\le N} f(n^2+1)$, showing almost-sure lower bounds matching the law of iterated logarithm, hence enriching the understanding of pseudo-random behavior in RMFs. These results extend Najnudel’s conjecture to the Rademacher setting and connect to broader Chowla-type questions for RMFs.
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 $κ_P>0$ such that \[ \frac{1}{\sqrt{κ_P N}}\sum_{n\leq N}f(P(n))\xrightarrow{d}\mathcal{N}(0,1), \] as $N\rightarrow\infty$, 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.