Giving a sense to the formal equation $\sin x=-\pi\sum_{n=1}^{+\infty}\frac{\mu(n)}n\left\{\frac{nx}{2\pi}\right\}$

Question

Does there exist a formula giving a sense to the formal equation $$ \sin x=-\pi\sum_{n=1}^{+\infty}\frac{\mu(n)}{n}\left\{\frac{nx}{2\pi}\right\}, $$ where $\mu$ is the Möbius function, $\{\cdot\}$ stands for the fractional part of a real number?

Namely, the series on the right hand side does not converge, but can it be made convergent to $\sin x$ after applying some "natural" summation method?

Answer 1

The series on the right-hand side does converge! (H. Davenport, On some infinite series involving arithmetical functions, Quart. J. Math. Oxford Ser. 8 (1937), 8-13.)