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

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