Bounded Exponential Sums with Multiplicative Coefficients

2025/06/15 by Pierre-Alexandre Bazin, Pylaiev, Ihor, Ihor Pylaiev +2 · 1 citation
Mathematics · #math.NT

paper · pdf · doi:10.48550/arxiv.2506.12845

Abstract

We investigate when the exponential sum S_f(x,α) := ∑_{n≤ x}f(n)e(nα) is bounded, for a multiplicative function f and α∈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.

Citations

• Bounds for exponential sums with random multiplicative coefficients
• Beyond the Erdős discrepancy problem in function fields
• Complex valued multiplicative functions with bounded partial sums
• Exponential sums with multiplicative coefficients and applications
• Exponential sums with multiplicative coefficients and applications
• A note on bounded exponential sums
• Multiplicative functions that are close to their mean
• The Erdős discrepancy problem over the squarefree and cubefree integers
• On the Duffin-Schaeffer conjecture
• Rigidity Theorems for Multiplicative Functions
• Rigidity theorems for multiplicative functions
• Correlations of multiplicative functions and applications
• The logarithmically averaged Chowla and Elliott conjectures for two-point correlations
• Multiplicative functions in short intervals
• On the Erdos Discrepancy Problem
• On Multiplicative Arithmetical Functions whose Modulus does not Exceed One
• Exponential sums with multiplicative coefficients
• Khintchine’s problem in metric Diophantine approximation

Cited by

• Exponential sums over primes are unbounded

Related

• L¹ means of exponential sums with multiplicative coefficients. II
• A note on bounded exponential sums
• Incomplete exponential sums over exponential functions
• Exponential sums twisted by general arithmetic functions
• Multiplicative mimicry and improvements of the Polya-Vinogradov\n inequality