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Almost sure bounds for a weighted Steinhaus random multiplicative function

Seth Hardy

TL;DR

The paper analyzes almost-sure fluctuations of the weighted partial sums $M_f(t)=\sum_{n\le t} f(n)/\sqrt{n}$ for a Steinhaus random multiplicative function, establishing sharp upper and lower bounds that align with a law-of-the-iterated-logarithm profile. The main results show $M_f(x) \ll \exp((1+\varepsilon)\sqrt{\log_2 x\ \log_4 x})$ a.s. and a matching lower bound in the $\limsup$ sense, reflecting the central role of the Euler product in dictating large fluctuations. The approach blends analytic and probabilistic techniques: decomposing $M_f$ into main and error terms guided by Euler products, conditioning on primes within dyadic scales, and applying harmonic-analysis identities to extract Euler-product sizes; probabilistic tools such as Lévy inequality, Berry–Esseen, and Borel–Cantelli forces yield sharp a.s. control. By closely tying $M_f(t)$ to its Euler product, the work substantiates the Euler-product model for zeta-type fluctuations on the critical line and provides a precise, almost-sure quantification of large fluctuations with clear implications for related random multiplicative models.

Abstract

We obtain almost sure bounds for the weighted sum $\sum_{n \leq t} \frac{f(n)}{\sqrt{n}}$, where $f(n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated logarithm, giving sharp upper and lower bounds.

Almost sure bounds for a weighted Steinhaus random multiplicative function

TL;DR

The paper analyzes almost-sure fluctuations of the weighted partial sums for a Steinhaus random multiplicative function, establishing sharp upper and lower bounds that align with a law-of-the-iterated-logarithm profile. The main results show a.s. and a matching lower bound in the sense, reflecting the central role of the Euler product in dictating large fluctuations. The approach blends analytic and probabilistic techniques: decomposing into main and error terms guided by Euler products, conditioning on primes within dyadic scales, and applying harmonic-analysis identities to extract Euler-product sizes; probabilistic tools such as Lévy inequality, Berry–Esseen, and Borel–Cantelli forces yield sharp a.s. control. By closely tying to its Euler product, the work substantiates the Euler-product model for zeta-type fluctuations on the critical line and provides a precise, almost-sure quantification of large fluctuations with clear implications for related random multiplicative models.

Abstract

We obtain almost sure bounds for the weighted sum , where is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated logarithm, giving sharp upper and lower bounds.
Paper Structure (13 sections, 6 theorems, 115 equations)

This paper contains 13 sections, 6 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof of Theorem \ref{['T:1']}
  3. Outline of the proof of Theorem \ref{['T:2']}
  4. Upper bound
  5. Bounding variation between test points
  6. Bounding on test points
  7. Conditioning on likely events
  8. Bounding the main term $\mathcal{C}_{i,j}$
  9. Bounding the error term $\mathcal{D}_{i,j}$
  10. Completing the proof of Proposition \ref{['p:sumthenprop']}
  11. Law of the iterated logarithm-type bound for the Euler product
  12. Probability of complements of integral events are summable
  13. Lower Bound

Key Result

Theorem 1

For any $\varepsilon > 0$, we have almost surely.

Theorems & Definitions (14)

  • Theorem 1: Upper Bound
  • Theorem 2: Lower Bound
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 1
  • proof : Proof of Theorem \ref{['T:1']}, assuming Proposition \ref{['P:Mxibound']}
  • Remark 2.3.1
  • Proposition 2
  • ...and 4 more