# 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, $$mu geqslant 2$$and 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 $$mu geqslant 3$$which is very unlikely;

• convergence (in particular absolute convergence) of the series (1) would imply $$mu leqslant 3$$which 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?)