TITLE: Convergence of $\sum\limits_{k=2}^\infty \frac{|\sin kx|}{\log k}$
QUESTION [6 upvotes]: For what values of $x \in \mathbb{R}$ do the series 
$$ \sum\limits_{k=2}^\infty \frac{|\sin kx|}{\log k} $$
converge (and how do you prove the rest diverge)?
The series converge trivially at $x = n\pi$ where $n \in \mathbb{Z}$, but I'm not sure about any others.

REPLY [2 votes]: For the case $x \neq n\pi$, we have
$$|\sin(kx)|\geqslant \sin^2(kx)= \frac1{2}[1-\cos(2kx)]$$
and
$$\sum_{k=2}^n \frac{|\sin(kx)|}{\log k} \geqslant \frac1{2}\sum_{k=2}^n \frac{1}{\log k}-\frac1{2}\sum_{k=2}^n \frac{\cos(2kx)}{\log k}.$$
The series does not converge since
$$\sum_{k=2}^\infty \frac{1}{\log k} = \infty$$
and the second sum on the RHS converges by the Dirichlet test.