Finding on Convergence of the Flint Hills and Cookson Hills Series based on a Summation Formula of Adamchik and Srivastava involving the Riemann Zeta Function

Abstract

This article showcases significant progress in solving two renowned problems in the calculus of series: the Flint Hills and Cookson Hills series. For almost twenty years, a long-standing question has remained unanswered in regard to their convergence. Mainly, proving the convergence of the Flint Hills series would significantly impact the redefinition of the upper bound for the irrationality measure of the number π. One of the results presented in this article is that the Flint Hills series converges to 30.3144... which leads to a redefinition of the upper bound for the irrationality measure of π, specifically μ(π)≤ 2.5. This work proposes a transformation that solves the mystery of the Flint Hills and Cookson Hills series. It is based on a summation formula developed by mathematicians Adamchik and Srivastava. By leveraging a specialized series supported by the Riemann zeta function, this approach successfully transforms the original Flint Hills and Cookson Hills series into novel convergent versions with unique significance. The resulting sequences linked to these series are positive and bounded and satisfy convergence. Moreover, this article extends the Flint Hills series when the cosecant function has an arbitrary complex argument n+iβ, with i=√(-1), establishing a new series representation based on the polylogarithm 〖Li〗_3 (e^i2k), with k=1,2,3,…, e the Euler’s number, which bears resemblance to the famous integral of the Bose-Einstein distribution as a relevant finding. This is a never-seen-before link between the Flint Hills series and polylogarithms. Furthermore, a relationship between the Apéry constant and the Flint Hills and Cookson Hills series has been established. This article presents a significant breakthrough in the calculus of series by introducing a new method based on the Riemann Zeta function and logarithmical expressions derived from the Adamchik and Srivastava summation formula. The novel approach extends the analysis of convergence criteria for series, addressing ambiguous cases characterized by abrupt jumps. Thus, the Flint Hills series converges to 30.3144... and the Cookson Hills series to 42.9949... as proved in this article.