Estimate the sum of expansions for fractional part function

Question

Are there any estimates for the sum of Fourier series expansions for fractional part function?: \begin{equation} \sum_{x=1}^{n}\sum_{k=1}^{\infty}\frac{\sin(2\pi k \frac{n}{x})}{k}, \; n \in \mathbb{N}^{+} \end{equation}

Answer 1

Here is a partial answer. According to the article linked in your question, $$ \sum_{k=1}^{\infty}\frac{\sin(2\pi k \frac{n}{x})}{k}= \begin{cases} \pi\left(\frac{1}{2}-\left\{\frac{n}{x}\right\}\right),&\mathrm{if}\,x\nmid n, \\ 0,&\mathrm{if}\,x\mid n. \end{cases} $$ Therefore, $$ \sum_{x=1}^{n}\sum_{k=1}^{\infty}\frac{\sin(2\pi k \frac{n}{x})}{k}= \sum_{x=1,\, x\nmid n}^{n}\pi\left(\frac{1}{2}-\left\{\frac{n}{x}\right\}\right). $$