# For any $x\in\mathbb R$ and any positive integer $n$,$\$is $\left|\sum_{k=1}^n\frac{\sin{kx}}{k}\right|\le2\sqrt{\pi}$ true?

I'm interested in finding the min of constants $C$ such that $$\left|\sum_{k=1}^n\frac{\sin{kx}}{k}\right|\le C.$$

By using computer, I reached the following expectation:

$$\left|\sum_{k=1}^n\frac{\sin{kx}}{k}\right|\le2\sqrt{\pi}$$ for any $x\in\mathbb R$ and any positive integer $n$.

I can neither prove this nor find any counterexample even by using computer. If my expectation is true, then could you show me how to prove that? Also, please show me whether $2\sqrt{\pi}$ is the min of such $C$.

If it's not true, please show me the counterexample. I need your help.

The best it is possible to state is: $$\left|\sum_{n=1}^N \frac{\sin(nx)}{n}\right|\leq\int_{0}^{1}\frac{\sin(\pi x)}{x}\,dx = 1.85194\ldots$$ Call $f_N(x)=\sum_{n=1}^N \frac{\sin(nx)}{n}$: it is a $2\pi$-periodic function converging to $\frac{\pi-x}{2}$ in $L_2\left([0,2\pi]\right)$. Since: $$\frac{d f_N(x)}{dx}= \frac{\cos\left(\frac{N+1}{2}x\right)\sin\left(\frac{N}{2}x\right)}{\sin\left(\frac{x}{2}\right)},$$ we know that $f_N(x)$ has $2n$ stationary points in $[0,2\pi]$, local maxima in $x=\frac{(2k+1)\pi}{N+1}$, the first one occurring in $x_N=\frac{\pi}{N+1}$. Once we prove that the value of $f_N(x)$ in any other local maximum is less than $f_N(x_N)$, and that $f_N(x_N)$ is an increasing sequence (I still must find a convincing proof of this two facts, but they look not too hard to deal with and strongly supported by computer inspection) the best bound we can hope in is: $$\left|\sum_{n=1}^N \frac{\sin(nx)}{n}\right|\leq\lim_{N\to +\infty}\sum_{n=1}^{N}\frac{\sin\left(\frac{\pi x}{N+1}\right)}{n},$$ where the RHS a Riemann sum associated with: $$\int_{0}^{1}\frac{\sin(\pi x)}{x}\,dx = 1.85194\ldots<\frac{13}{7},$$ QED.