论文标题
非理性措施和Flint-Hills系列的界限
Bounds on Irrationality Measures and the Flint-Hills Series
论文作者
论文摘要
未知是否Flint-Hills系列$ \ sum_ {n = 1}^\ infty \ frac {1} {n^3 \ sin^2（n）} $收敛。 Alekseyev（2011）将这个问题与$π$的非理性度量相关联，即$μ（π）> \ frac {5} {2} $将暗示Flint-Hills系列的分歧。在本文中，我们建立了一个近乎完整的匡威，那就是$μ（π）<\ frac {5} {2} $将暗示收敛。关于密切合理近似密度的相关结果可能具有独立的关注。简要解决了$μ（π）= \ frac {5} {2} $的剩余边缘情况，并证明很难解决。
It is unknown whether the Flint-Hills series $\sum_{n=1}^\infty \frac{1}{n^3\sin^2(n)}$ converges. Alekseyev (2011) connected this question to the irrationality measure of $π$, that $μ(π) > \frac{5}{2}$ would imply divergence of the Flint-Hills series. In this paper we established a near-complete converse, that $μ(π) < \frac{5}{2}$ would imply convergence. The associated results on the density of close rational approximations may be of independent interest. The remaining edge case of $μ(π) = \frac{5}{2}$ is briefly addressed, with evidence that it would be hard to resolve.
