convergence of series with parameters

Question

I need to determine for which pairs $\theta, \beta \in \mathbb{R}$ the following series converges: $\sum ^{\infty }_{n=1}\dfrac{\cos \left( n\theta \right) }{n^{\beta}}$.
At first, for any $\beta >1$, we have that $\left| \dfrac{\cos \left( n\theta \right) }{n^\beta }\right| \leq \dfrac{1}{n^{\beta}}$, so from the comparison test for any $\beta > 1$ and for every $\theta \in \mathbb{R}$ we get that the series is absolutely convergent, hence the series is convergent.
As for other pairs I can't find any tests that help me.

Answer 1

Some easy cases.

$\theta=2\pi \implies$ diverges for $\beta=1$.

$\theta=\pi \implies$ converges for $\beta=1$.

Not sure what happens for $\theta$ a rational multiple of $\pi$.

Have no idea for $\theta$ an irrational multiple of $\pi$.