Order of Magnitude of Fourier Coefficients for Almost Periodic Functions

TL;DR

This paper explores the behavior of Fourier coefficients for almost periodic functions, extending classical results and applying these insights to the Riemann-Zeta function, highlighting new theoretical connections.

Contribution

It generalizes Fourier coefficient estimates from periodic to almost periodic functions and applies these results to the Riemann-Zeta function.

Findings

Fourier coefficients for almost periodic functions are bounded similarly to the periodic case.

A new estimate for Fourier coefficients is proved for almost periodic functions.

Application of the estimate provides insights into the Riemann-Zeta function.

Abstract

We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then provide an application of our estimate to the Riemann-Zeta Function.