关于Flint Hills级数的收敛性
On convergence of the Flint Hills series
摘要：
尚不清楚Flint Hills级数$\sum_{n=1}^{\infty} \frac{1}{n^3\cdot\sin(n)^2}$是否收敛。我们发现，这个问题是密切相关的$π$，表示$μ(π)$的非理性措施。特别是，收敛的弗林特山系列将意味着$μ(π) \leq 2.5$这是远远强于目前已知的最好的上界$μ(π)\leq 7.6063...$。这个结果很容易推广到$\sum_{n=1}^{\infty} \frac{1}{n^u\cdot |\sin(n)|^v}$形式的级数，其中$u,v>0$。我们使用目前已知的$μ(π)$的界限推导出条件$u$和$v$，保证收敛的系列。
英文摘要：
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 $π$, denoted $μ(π)$. In particular, convergence of the Flint Hills series would imply $μ(π) \leq 2.5$ which is much stronger than the best currently known upper bound $μ(π)\leq 7.6063...$. 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 $μ(π)$ to derive conditions on $u$ and $v$ that guarantee convergence of such series.
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