The Flint Hills Series, Mixed Tate Motives, and a Criterion for the Irrationality Measure of $π$

Abstract

We undertake a rigorous structural analysis of the Flint Hills series $S = \sum_{n=1}^{\infty} \frac{1}{n^3 \sin^2 n}$. Our primary contribution is a reduction theorem that expresses $S$ as a linear combination of $ζ(3)$ and a companion series $R_1^* = \sum_{n=1}^\infty \frac{\sin 3n}{n^3 \sin^3 n}$, with the equivalence "$S$ converges if and only if $R_1^*$ converges" holding unconditionally.We prove that this equivalence, combined with the classical result of Alekseyev, yields a sharp arithmetic criterion: $S$ converges if and only if the irrationality measure $μ(π) \leq 5/2$. Conditionally on this bound, we identify $R_1^*$ as a period of a Mixed Tate Motive of weight 3 over the ring of integers $O_K$ of the imaginary quadratic field $K = Q(\sqrt{-3})$, lying in the image of the Borel regulator on $K_5(O_K)$. This gives a precise conjectural closed form for $S$ as a $Q$-linear combination of $ζ(3)$ and $L(3, χ_{-3})$ modulo a geometric correction term. All analytic identities are verified to fifty decimal places of precision.

Theorems & Definitions (29)

• Theorem 1.1: Alekseyev, 2011
• Theorem 1.2: Trigonometric Reduction
• Theorem 1.3: Sharp Arithmetic Criterion
• Theorem 1.4: Conditional Motivic Identity
• Remark 1.5
• Lemma 2.1: Triple-angle decomposition
• proof
• Lemma 2.2: Linear inversion
• proof
• proof : Proof of Theorem \ref{['thm:reduction']}