# Signed variant of the Flint Hills series

I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. 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 convergence of this series been studied in literature?

My immediate answer was that this must have something to do with the irrationality measure $$\mu$$ of $$\pi$$. Obviously, $$\mu \geqslant 2$$, and the best currently known upper bound for $$\mu$$ is $$\mu \leqslant 7.6063\!\ldots\,$$, due to Salikhov; see [V. Kh. Salikhov, On the Irrationality Measure of $$\pi$$. Russ. Math. Surv 63(3):570–572, 2008]. It is widely believed that $$\mu = 2$$.

In fact, the inequality $$\mu < 3$$ is equivalent to convergence of $$1 / (n^2 \sin n)$$ to zero. Thus 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), then one could easily show that $$1 / (n^2 \sin n) = O(n^{-1 - \varepsilon})$$ for some $$\varepsilon > 0$$, which would imply absolute convergence of the series (1).

My student searched the web and realized that his question is related to the well-known open problem, asking whether the Flint Hills series $$\sum_{n = 1}^\infty \frac{1}{n^3 \sin^2 n} \tag{2}$$ converges. An extension of this problem asks for convergence of a more general series $$\sum_{n = 1}^\infty \frac{1}{n^p |\sin n|^q} , \tag{3}$$ which is equivalent to absolute convergence of the series (1) when $$p = 2$$ and $$q = 1$$. For more details, see [Max. A. Alexeyev, On convergence of the Flint Hills series, arXiv:1104.5100, 2011].

To summarise, this 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 it is certainly an open problem;

• lack of absolute convergence of the series (1) would not have any consequences for the estimates of $$\mu$$.

Question (long version)

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

Edited: I just noticed David Simmons's answer to an MO question on the Flint Hills series, which reduces the question of its convergence to a similar question for a series involving convergents of $$\pi$$. The same argument should work for the absolute convergence of the series (1), but I do not see right away if it leads to an answer to question 1.

You mentioned my argument from the other thread, which can be used to show that if $$\mu(\pi) < 3$$, then your series converges. But this also means you can't get any further information about $$\mu(\pi)$$ from knowing that your series converges -- it would converge if $$\pi$$ were any of the numbers with exponent of irrationality $$<3$$, so you cannot rule out $$\pi$$ being any of those numbers (I know I am speaking of $$\pi$$ as though it were a variable, but this is necessary in order for questions like "does A imply B?" to make sense when A and B are fixed) In other words, the answer to question 1 is "No".