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Bounded Exponential Sums with Multiplicative Coefficients

Pierre-Alexandre Bazin, Ihor Pylaiev, Fred Tyrrell

Abstract

We investigate when the exponential sum $S_f(x,α) := \sum_{n\le x}f(n)\mathrm{e}(nα)$ is bounded, for a multiplicative function $f$ and $α\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,α)$ is bounded only when $f$ is very close to a twisted Dirichlet character $χ(n)n^{it}$. We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when $f$ is completely multiplicative and $α$ is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of $α$.

Bounded Exponential Sums with Multiplicative Coefficients

Abstract

We investigate when the exponential sum is bounded, for a multiplicative function and . We show that under natural assumptions, is bounded only when is very close to a twisted Dirichlet character . We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when is completely multiplicative and is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of .

Paper Structure

This paper contains 18 sections, 22 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Discussion of results
  5. Overview of the paper
  6. Reduction to pretentious functions
  7. Dirichlet characters
  8. Pretentious functions
  9. Performing the reduction
  10. The rotation trick
  11. Lemmata for the rotation trick
  12. Applying the rotation trick
  13. A result depending on $\alpha$
  14. Statement of results
  15. Technical lemmas
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $f: \mathbb{N} \rightarrow \mathbb{U} \cup \{0\}$ be a multiplicative function, and assume that If, for some $\alpha \in \mathbb{R}$, then there is a Dirichlet character $\chi$ and a real number $t \in \mathbb{R}$ such that

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 38 more