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Convergence of the Flint Hills series

Corollary 1.2: $\sum_{n\ge1}1/(n^3\sin^2 n)$ converges, deduced from $\mu(\pi)=2$ via a dyadic spacing estimate valid for any exponent below $5/2$ . More generally $\sum n^{-a}|\sin n|^{-b}$ converges iff $a>\max\{1,b\}$ (Corollary 5.2). No value or numerical estimate of the sum is given.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Convergence of the Flint Hills series · Flint Hills series

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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Attribution

VibeMathed
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Unreleased internal OpenAI model
model · ai model contributor · OpenAI

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Named evidence

VibeMathed record: Convergence of the Flint Hills series

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Corollary 1.2 and Section 5 of the TeX source read against the question; convergence follows from $\mu(\pi)<5/2$ by a short dyadic spacing lemma that the paper proves (Meiburg's criterion is cited too). Everything rests on the unrefereed main theorem $\mu(\pi)=2$ . The release's Lean scope note says the Flint Hills consequence is outside the Comparator challenge PiExponent. The solution module OAI.NumberTheory.PiExponent.Main also contains a theorem flint_hills_summable stating summability of $1/(n^3\sin^2 n)$ , but it is not a Comparator challenge, so it is not counted here; nothing was rebuilt.

Source ↗

The irrationality exponent of pi is 2

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Alekseyev, On convergence of the Flint Hills series (arXiv 2011)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Abridged reasoning summary released by OpenAI for this family

witness · pending

other link supplied by the VibeMathed registry.

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This event attributed to Unreleased internal OpenAI model

Convergence of the Flint Hills series parent of this event

The irrationality exponent of pi is 2 evidence for this event

VibeMathed record: Convergence of the Flint Hills series evidence for this event

Alekseyev, On convergence of the Flint Hills series (arXiv 2011) evidence for this event

Abridged reasoning summary released by OpenAI for this family evidence for this event

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