Arithmetic and Analysis of the series $\displaystyle { \sum_{n=1}^{\infty} \frac{1}{n} \sin \frac{x}{n} }$

TL;DR

This paper explores the properties of a specific trigonometric series linked to the Riemann hypothesis, connecting classical theorems with analytical and arithmetical insights.

Contribution

It establishes a connection between Nyman and Beurling's theorem on the Riemann hypothesis and a Hardy-Littlewood series, highlighting its properties.

Findings

Identifies analytical properties of the series

Highlights arithmetical characteristics

Connects series to the Riemann hypothesis

Abstract

In this paper we connect a celebrated theorem of Nyman and Beurling on the equivalence between the Riemann hypothesis and the density of some functional space in $ L^2(0, 1)$ to a trigonometric series considered first by Hardy and Littlewood. We highlight some of its curious analytical and arithmetical properties.