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

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.