Paper

{
  "raw_xml": "<entry>\n    <id>http://arxiv.org/abs/2603.09719v1</id>\n    <title>The Flint Hills Series, Mixed Tate Motives, and a Criterion for the Irrationality Measure of $π$</title>\n    <updated>2026-03-10T14:27:48Z</updated>\n    <link href='https://arxiv.org/abs/2603.09719v1' rel='alternate' type='text/html'/>\n    <link href='https://arxiv.org/pdf/2603.09719v1' rel='related' title='pdf' type='application/pdf'/>\n    <summary>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.</summary>\n    <category scheme='http://arxiv.org/schemas/atom' term='math.NT'/>\n    <published>2026-03-10T14:27:48Z</published>\n    <arxiv:comment>10 pages. Includes a motivic interpretation of the Flint Hills series and numerical verification to 50 decimal places</arxiv:comment>\n    <arxiv:primary_category term='math.NT'/>\n    <author>\n      <name>Carlos Lopez Zapata</name>\n    </author>\n  </entry>"
}