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 (see (3.4)). We highlight some of its curious analytical and arithmetical properties.

中文翻译：

---

级数的算术和分析$$ {\ sum _ {n = 1} ^ {\ infty} \ frac {1} {n} \ sin \ frac {x} {n}} $$ ∑ n = 1∞1 n罪恶

在本文中，我们将著名的Nyman和Beurling定理与Riemann假设和\（L ^ 2（0，1）\）中某些功能空间的密度之间的等价关系与Hardy和Littlewood（见（3.4））。我们重点介绍了它的一些好奇的分析和算术性质。