[Paper Review] On convergence of the Flint Hills series

Max A. Alekseyev|arXiv (Cornell University)|Apr 27, 2011

Meromorphic and Entire Functions11 references3 citations

TL;DR

This paper establishes a direct link between the convergence of the Flint Hills series ∑1/(n³·sin²(n)) and the irrationality measure μ(π) of π. It proves that convergence of the series would imply μ(π) ≤ 2.5, significantly improving the current best upper bound of 7.6063. The result generalizes to series of the form ∑1/(nᵘ·|sin(n)|ᵛ), with convergence conditions derived from μ(π).

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 $π$, 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.

Motivation & Objective

• To investigate the convergence of the Flint Hills series ∑₁^∞ 1/(n³·sin²(n))
• To establish a connection between the convergence of this series and the irrationality measure μ(π) of π
• To generalize the convergence conditions to broader classes of series ∑1/(nᵘ·|sin(n)|ᵛ) for u,v > 0
• To derive convergence criteria based on known upper bounds for μ(π)
• To clarify the implications of convergence or divergence for the value of μ(π)

Proposed method

• Uses the definition of the irrationality measure μ(π) as the infimum of m such that |π - p/q| < 1/qᵐ holds for only finitely many coprime p,q
• Applies Lemma 1 to bound |sin(n)| from below using the distance from n to the nearest multiple of π
• Relies on the inequality |sin(n)| ≥ (2/π)·|n - mπ| for |n - mπ| ≤ π/2, where m = ⌊n/π⌋
• Uses the fact that if |π - p/q| < 1/qᵏ for only finitely many q, then |sin(n)| grows at least as fast as n⁻ᵏ⁺¹ for large n
• Applies the bound 1/(nᵘ·|sin(n)|ᵛ) = O(1/nᵘ⁻⁽μ(π)⁻¹⁾ᵛ⁻ᵉ) for any ε > 0 to analyze decay rate
• Applies the p-series convergence test by comparing to ζ(w) with w = u - v(μ(π) - 1) - ε > 1

Experimental results

Research questions

• RQ1 Does the Flint Hills series ∑₁^∞ 1/(n³·sin²(n)) converge?
• RQ2 What is the relationship between the convergence of the Flint Hills series and the irrationality measure μ(π)?
• RQ3 Under what conditions on u and v does the generalized series ∑₁^∞ 1/(nᵘ·|sin(n)|ᵛ) converge?
• RQ4 How do known upper bounds for μ(π) affect the convergence of such series?
• RQ5 Can the divergence of the series imply a lower bound on μ(π)?

Key findings

• Convergence of the Flint Hills series ∑₁^∞ 1/(n³·sin²(n)) would imply that μ(π) ≤ 2.5, a significantly stronger bound than the current best-known upper bound of 7.6063.
• If μ(π) < 1 + u/v, then the sequence 1/(nᵘ·|sin(n)|ᵛ) converges to zero.
• If μ(π) > 1 + u/v, then the sequence 1/(nᵘ·|sin(n)|ᵛ) diverges, as it contains a subsequence bounded away from zero.
• The series ∑₁^∞ 1/(nᵘ·|sin(n)|ᵛ) converges whenever μ(π) < 1 + (u - 1)/v, which includes cases like (u,v) = (8,1), (15,2), and (21,3) under the current bound μ(π) ≤ 7.6063.
• The divergence of the Flint Hills series would not yield a non-trivial lower bound on μ(π), as per Corollary 6.
• The best-known upper bound μ(π) ≤ 7.6063 implies that the sequence 1/(nᵘ·|sin(n)|ᵛ) converges to zero for all (u,v) such that 1 + u/v > 7.6063, including (7,1), (14,2), and (20,3)