Find the set of $k \in \mathbb{R}$ such that:
$$\sum_{n=1}^{\infty}\frac{(-1)^n\sin{(k\log{n})}}{\sqrt{n}}=0$$
I understand that this is a convergent Dirichlet series, but I can't quite wrap my head around how this type of problem can be approached. For further context, this series originates from the comments of an older post on converging Dirichlet series, found here.