Convergence of $\sum(n^p\sin^qn)^{-1}$

Question

I've been recently interested in the problem of convergence of the function in such form: $\displaystyle \sum_{n=1}^\infty\frac1{n^p\sin^q n}$.

I saw there's been discussion here when $p=3, q=2$ and $p=2, q=1$ , but wondering if there's any advancement in other cases of variant of the Flint Hills series such as $p=3, q=1$

(and also this post also helped my thought)

any relevant paper, article or suggestion will be appreciated. Thanks.

Answer 1

A while ago I've addressed this question in On convergence of the Flint Hills series.