In one exercise of the Calculus textbook, Thomas' Calculus, 13th edition, there is an exercise (Section 10.4, 71) says it is not known yet if the series $$\sum_{n=1}^\infty \frac{1}{n^3 \sin^2(n)} $$ is convergent or not. The reason should be: given a fraction $p/q$ which is close to $\pi$, the term $$\frac{1}{p^3 \sin^2(p)}$$ can be large, and it should be very hard to estimate of the contributions of such terms. For example, if $p/q=355/133$, the corresponding term $$\frac{1}{355^3 \sin^2(355)}=24.598$$ and thus the partial sum has a jump at 355.
I have the following questions:
Question 1, Do we know anything new about this series?
Question 2: Do we know convergence/divergence about similar series, like $$\sum_{n=1}^\infty \frac{1}{n^\alpha \sin^2(n)}, \alpha>1?$$ Is there something special about the power $\alpha=3$ as in the exercise?
Thanks in advance.
Edit: Thanks @octave for the reference https://arxiv.org/pdf/1104.5100.pdf, this series is called Flint Hills series. I just found the link https://mathoverflow.net/questions/24579/convergence-of-sumn3-sin2n-1/172417#172417 in math overflow on the same question.