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Bounds for exponential sums with random multiplicative coefficients

Seth Hardy

TL;DR

This work resolves the asymptotic size of the maximum of normalized exponential sums with random multiplicative coefficients, proving a sharp $\sqrt{\log N}$ lower bound for both Rademacher and Steinhaus models. It introduces a Gaussian-approximation framework for the lower bound, complemented by a delicate analysis of variances and covariances and a Normal-Comparison step to control maxima, while also establishing a partial upper bound in the Steinhaus case by exploiting conditional variances and a dyadic/ Brun–Titchmarsh argument. The methods blend probabilistic approximations, Diophantine-structure considerations, and classical exponential-sum bounds to handle contributions from large and small prime factors separately. The results significantly clarify the true order of magnitude for these random multiplicative exponential sums and provide a pathway to tighter bounds via refined variance-control and Gaussian comparison techniques. Overall, the paper advances understanding of how multiplicative randomness interacts with additive harmonic sums, with potential implications for related random multiplicative models and their extremal behavior.

Abstract

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$ \max_{θ\in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n θ) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N \rightarrow \infty$. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon > 0$, we prove the partial upper bound result $$ \max_{θ\in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n θ) \Bigr| \ll {(\log N)}^{7/4 + \varepsilon},$$ asymptotically almost surely as $N \rightarrow \infty$, where $P(n)$ denotes the largest prime factor of $n$.

Bounds for exponential sums with random multiplicative coefficients

TL;DR

This work resolves the asymptotic size of the maximum of normalized exponential sums with random multiplicative coefficients, proving a sharp lower bound for both Rademacher and Steinhaus models. It introduces a Gaussian-approximation framework for the lower bound, complemented by a delicate analysis of variances and covariances and a Normal-Comparison step to control maxima, while also establishing a partial upper bound in the Steinhaus case by exploiting conditional variances and a dyadic/ Brun–Titchmarsh argument. The methods blend probabilistic approximations, Diophantine-structure considerations, and classical exponential-sum bounds to handle contributions from large and small prime factors separately. The results significantly clarify the true order of magnitude for these random multiplicative exponential sums and provide a pathway to tighter bounds via refined variance-control and Gaussian comparison techniques. Overall, the paper advances understanding of how multiplicative randomness interacts with additive harmonic sums, with potential implications for related random multiplicative models and their extremal behavior.

Abstract

For a Rademacher or Steinhaus random multiplicative function, we prove that asymptotically almost surely as . Furthermore, for a Steinhaus random multiplicative function, and any , we prove the partial upper bound result asymptotically almost surely as , where denotes the largest prime factor of .
Paper Structure (13 sections, 6 theorems, 99 equations)

This paper contains 13 sections, 6 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Heuristic discussion
  4. Outline of the proof of the lower bound, Theorem \ref{['T:lb']}
  5. Outline of the proof of the upper bound, Theorem \ref{['T:ub']}
  6. Lower Bound
  7. Approximating by a normal
  8. The 'big Oh' terms are small
  9. Variance and covariance estimates
  10. Combining results
  11. Upper Bound
  12. Conditional variance bound
  13. Completion of the proof of Theorem \ref{['T:ub']}

Key Result

Theorem 1

Let $f$ be a Rademacher or Steinhaus random multiplicative function. We have with probability $1 - o(1)$ as $N \rightarrow \infty$. In the Rademacher and Steinhaus cases, we calculate that the implicit constants can be taken to be $\frac{4 \sqrt{6}}{29 \pi}$ and $\frac{4}{29}$ respectively.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Remark
  • Lemma 1
  • proof
  • Remark
  • proof
  • Remark 1
  • proof : Proof of Theorem \ref{['T:lb']}.
  • Proposition 1
  • ...and 8 more