Central limit theorems for random multiplicative functions
Kannan Soundararajan, Max Wenqiang Xu
TL;DR
This work advances the understanding of central limit phenomena for random multiplicative functions by proving a general criterion that yields a CLT when sums are restricted to dense, energy-controlled subsets ${\mathcal A}$. Central to the approach is a quantitative martingale CLT that handles complex-valued sums via a decomposition along primes and a subset ${\mathcal S}$ with small multiplicative energy $E_×(\mathcal S)$. The paper applies this framework to a variety of settings: short-interval sums, sums of two squares, shifted primes, and sums twisted by irrational phases $e(n\theta)$ under Diophantine conditions, obtaining Gaussian limits in ranges beyond previous thresholds. Moreover, it extends the program to large model subsets arising from Ford’s multiplication-table ideas and outlines parallel results for the Rademacher model with an appropriate square-energy notion. Collectively, the results illuminate when and why “more than square-root cancellation” still allows Gaussian fluctuations after suitable restriction, with connections to sieve methods, additive combinatorics, and Diophantine analysis.
Abstract
A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that $\sum_{n\le N} f(n)$ exhibits ``more than square-root cancellation," and in particular $\frac 1{\sqrt{N}} \sum_{n\le N} f(n)$ does not have a (complex) Gaussian distribution. This paper studies $\sum_{n\in {\mathcal A}} f(n)$, where ${\mathcal A}$ is a subset of the integers in $[1,N]$, and produces several new examples of sets ${\mathcal A}$ where a central limit theorem can be established. We also consider more general sums such as $\sum_{n\le N} f(n) e^{2πi nθ}$, where we show that a central limit theorem holds for any irrational $θ$ that does not have extremely good Diophantine approximations.