Number Theory – Signed variant of the Flint Hills series

I asked my Calculus 2 students to propose a series that they could not comment on. One of the students, Denis Zelent, invented a very interesting one:
$$
sum_ {n = 1} ^ infty frac {1} {n ^ 2 sin n} ,. tag {1}
$$

Question (short version): Has the convergence of this series been studied in the literature?

My immediate answer was that it must have something to do with the measure of irrationality $ mu $ of $ pi $. obviously, $ mu geqslant $ 2and the best known upper limit for $ mu $ is $ mu leqslant 7.6063 ! ldots , $, due to Salikhov; see[VKhSalikhov[VKhSalikhov[VKhSalikhov[VKhSalikhovOn the measure of irrationality of $ pi $. Russ. Math. Surv 63 (3): 570-572, 2008]. It is widely accepted that $ mu = $ 2.

In fact, inequality $ mu <$ 3 is equivalent to the convergence of $ 1 / (n ^ 2 sin n) $ to zero. So if we knew that $ mu geqslant $ 3, the series (1) would diverge. On the other hand, if we had $ mu <$ 2 (which is of course absurd), so we could easily show that $ 1 / (n ^ 2 sin n) = O (n ^ {- 1 – varepsilon}) $ for some people $ varepsilon> $ 0, which would imply an absolute convergence of the series (1).

My student did a web search and realized that his question was related to the well-known open problem, asking if the Flint Hills Series
$$
sum_ {n = 1} ^ infty frac {1} {n ^ 3 sin ^ 2 n} tag {2}
$$

converges. An extension of this problem requires the convergence of a more general series
$$
sum_ {n = 1} ^ infty frac {1} {n ^ p | sin n | ^ q}, tag {3}
$$

which is equivalent to the absolute convergence of the series (1) when $ p = $ 2 and $ q = $ 1. For more details, see[MaxAAlexeyev[MaxAAlexeyev[MaxAAlexeyev[MaxAAlexeyevOn the convergence of the Flint Hills series, arXiv: 1104.5100, 2011].

To summarize, here is what we have found so far:

  • lack of convergence of $ 1 / (n ^ 2 sin n) $ to zero would imply that $ mu geqslant $ 3which is very unlikely;

  • convergence (in particular absolute convergence) of the series (1) would imply $ mu leqslant $ 3which means that it is certainly an open problem;

  • The absence of absolute convergence of the series (1) would have no impact on the estimates of $ mu $.

Question (long version)

  1. Does the absolute convergence of the series (1) imply narrower limits on irrationality measure estimates? $ mu $ of $ pi $?
  2. Vice versa: assuming that $ mu $ is known, can we say if the series (1) converges absolutely?
  3. Same questions with absolute convergence changed into convergence. In other words: are there any help cancellations here?
  4. Does the series (1) have a fancy name similar to the Flint Hills and Cookson Hills series? (And if not, can Denis choose an appropriate mountain range?)