2022/08/29 by Meiburg, Alex · 1 citation
#11J82 #40A05 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2208.13356
It is unknown whether the Flint-Hills series ∑n=1^∞ (1)/(n3sin2(n)) converges. Alekseyev (2011) connected this question to the irrationality measure of π, that μ(π) > (5)/(2) would imply divergence of the Flint-Hills series. In this paper we established a near-complete converse, that μ(π) < (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) = (5)/(2) is briefly addressed, with evidence that it would be hard to resolve.