On convergence of the Flint Hills series

Abstract

It is not known whether the Flint Hills series $\sum_{n=1}^{\infty} \frac{1}{n^3\cdot\sin(n)^2}$ converges. We show that this question is closely related to the irrationality measure of $\pi$, denoted $\mu(\pi)$. In particular, convergence of the Flint Hills series would imply $\mu(\pi) \leq 2.5$ which is much stronger than the best currently known upper bound $\mu(\pi)\leq 7.6063...$. This result easily generalizes to series of the form $\sum_{n=1}^{\infty} \frac{1}{n^u\cdot |\sin(n)|^v}$ where $u,v>0$. We use the currently known bound for $\mu(\pi)$ to derive conditions on $u$ and $v$ that guarantee convergence of such series.