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Almost sure bounds for weighted sums of Rademacher random multiplicative functions

Christopher Atherfold

TL;DR

This work analyzes almost-sure bounds for the weighted sums $M_f(x)=\sum_{n\le x} \frac{f(n)}{\sqrt{n}}$ where $f$ is a Rademacher random multiplicative function. Building on Caich’s and Hardy’s methods, the authors reduce to evaluation on a sparse grid of test points, decompose by prime factors, and control fluctuations via random Euler products and multiplicative-chaos techniques (notably Harper’s low-moments results). They prove an almost-sure upper bound $M_f(x) \ll (\log\log x)^{61/80+\varepsilon}$ and a corresponding lower-bound fluctuation bound $(\log\log x)^{-1/2}$, with a sharper bound available when restricting to integers with a large prime factor, namely $\sum_{\substack{n\le x\\ P(n)>\sqrt{x}}}\frac{f(n)}{\sqrt{n}} \ll (\log\log x)^{21/80+\varepsilon}$. The analysis highlights the role of multiplicative-chaos contributions in the Rademacher model and contrasts with the Steinhaus case, showing a larger- than-expected fluctuation behavior driven by random Euler products and Gaussian-walk phenomena. The results deepen the understanding of how random multiplicative models reflect large fluctuations and contribute to the broader program connecting random multiplicative functions with zeta-function heuristics.

Abstract

We prove that when $f$ is a Rademacher random multiplicative function for any $ε>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{61/80+ε}$ for almost all $f$. We also show that there exist arbitrarily large values of $x$ such that $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \gg (\log\log(x))^{-1/2}$. This is different to what is found in the Steinhaus case, this time with the size of the Rademacher Euler product making the multiplicative chaos contribution the dominant one. We also find a sharper upper bound when we restrict to integers with a prime factor greater than $\sqrt{x}$, proving that $\sum_{\substack{n \leqslant x \\ P(n) > \sqrt{x}}}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{21/80+ε}$.

Almost sure bounds for weighted sums of Rademacher random multiplicative functions

TL;DR

This work analyzes almost-sure bounds for the weighted sums where is a Rademacher random multiplicative function. Building on Caich’s and Hardy’s methods, the authors reduce to evaluation on a sparse grid of test points, decompose by prime factors, and control fluctuations via random Euler products and multiplicative-chaos techniques (notably Harper’s low-moments results). They prove an almost-sure upper bound and a corresponding lower-bound fluctuation bound , with a sharper bound available when restricting to integers with a large prime factor, namely . The analysis highlights the role of multiplicative-chaos contributions in the Rademacher model and contrasts with the Steinhaus case, showing a larger- than-expected fluctuation behavior driven by random Euler products and Gaussian-walk phenomena. The results deepen the understanding of how random multiplicative models reflect large fluctuations and contribute to the broader program connecting random multiplicative functions with zeta-function heuristics.

Abstract

We prove that when is a Rademacher random multiplicative function for any , then for almost all . We also show that there exist arbitrarily large values of such that . This is different to what is found in the Steinhaus case, this time with the size of the Rademacher Euler product making the multiplicative chaos contribution the dominant one. We also find a sharper upper bound when we restrict to integers with a prime factor greater than , proving that .
Paper Structure (21 sections, 25 theorems, 219 equations)

This paper contains 21 sections, 25 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Ideas for Proofs
  3. Organisation of the paper and notation
  4. Preliminary results
  5. Upper bound
  6. Reduction to test points
  7. Splitting $M_f(x_i)$ by prime factors
  8. Decomposing the $S_{i,j}$
  9. Bounding the complement events
  10. Bounding $\mathcal{Q}_\ell^{(1)}$
  11. Preparing to apply Proposition \ref{['HA1']}
  12. Handling the good part of $\mathcal{Q}_\ell^{(1)}$
  13. Preparing to bound the bad part of $\mathcal{Q}_\ell^{(*)}$
  14. Bounding the bad part of $\mathcal{Q}_\ell^{(*)}$
  15. Bounding $\mathbb{P}(\mathcal{H}_\ell(T))$ and $\mathbb{P}(\mathcal{H}_\ell')$
  16. ...and 6 more sections

Key Result

Theorem 1

For any $\epsilon>0$ we have almost surely

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • proof
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • ...and 26 more