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The distribution of partial sums of random multiplicative functions with a large prime factor

Seth Hardy

TL;DR

This work analyzes the distribution of normalized partial sums of a Steinhaus random multiplicative function over integers with a large prime factor, proving that the subsum scaled by $\frac{(\log\log x)^{1/4}}{\sqrt{x}}$ converges to $\sqrt{C V_{\mathrm{crit}}}\,\mathcal{CN}(0,1)$, where $C=(e^{-\gamma}\log 2)/(2\pi)$ and $V_{\mathrm{crit}}$ is a GMC-derived random variable independent of the Gaussian limit. The authors connect the variance to mean-square Euler products and show that it concentrates on a GMC measure at the critical parameter, using a conditional Gaussian approximation and an extension of Saksman–Webb's GMC results to the full integral. They establish moment convergence for $0<q<2$, and provide sharp heavy-tail behavior for the scaled subsums, reflecting the GMC extremal structure. The results illuminate the non-Gaussian, heavy-tailed distribution of full partial sums and strengthen the bridge between random multiplicative functions and Gaussian multiplicative chaos, with implications for understanding the full sum distribution and related conjectures.

Abstract

For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.

The distribution of partial sums of random multiplicative functions with a large prime factor

TL;DR

This work analyzes the distribution of normalized partial sums of a Steinhaus random multiplicative function over integers with a large prime factor, proving that the subsum scaled by converges to , where and is a GMC-derived random variable independent of the Gaussian limit. The authors connect the variance to mean-square Euler products and show that it concentrates on a GMC measure at the critical parameter, using a conditional Gaussian approximation and an extension of Saksman–Webb's GMC results to the full integral. They establish moment convergence for , and provide sharp heavy-tail behavior for the scaled subsums, reflecting the GMC extremal structure. The results illuminate the non-Gaussian, heavy-tailed distribution of full partial sums and strengthen the bridge between random multiplicative functions and Gaussian multiplicative chaos, with implications for understanding the full sum distribution and related conjectures.

Abstract

For a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums where denotes the largest prime factor of . We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.

Paper Structure

This paper contains 20 sections, 25 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Previous work
  4. Key results
  5. Connection to Gaussian multiplicative chaos
  6. Outline of the proof of Theorem \ref{['t:variance']} and structure of the paper
  7. Preliminary number theory results
  8. Preliminary results on random multiplicative functions
  9. Expectation results
  10. Multiplicative Chaos Results
  11. Conditional normal approximation
  12. Analysis of the variance: proof of Theorem \ref{['t:variance']}
  13. Initial analysis
  14. Conditional concentration of the variance on small primes
  15. Distribution of the variance: proof of Theorem \ref{['t:SWext']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let $P(n)$ denote the largest prime factor of $n$, let $\gamma \coloneqq \lim_{n \rightarrow \infty} (\sum_{k=1}^n \frac{1}{k} - \log n )$ be the Euler--Mascheroni constant, and let $C = \frac{e^{- \gamma} \log 2}{2 \pi}$. For $f$ a Steinhaus random multiplicative function, we have where $\mathcal{CN} (0,1)$ is a standard complex Gaussian random variable and $V_{\mathrm{crit}}$ is a random variab

Theorems & Definitions (52)

  • Theorem 1
  • Corollary 1: Convergence of moments
  • Corollary 2: Heavy tails
  • Remark 1.1
  • Remark 1.2
  • Theorem 2
  • Theorem 3: Extension of Theorem 1.9 of SaksmanWebb
  • Lemma 3.1: Mean value over large primes
  • proof
  • Lemma 3.2: Pointwise bound for sums over primes
  • ...and 42 more