HEXAGONbeta

Irrational even beta values from powered hypergeometric kernels

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theory

Contributed by twinheterodox ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

For the Dirichlet beta function, we prove that at least one of β(6),β(8),…,β(74)\beta(6),\beta(8),\ldots,\beta(74) is irrational, and that at least one of β(8),β(10),…,β(110)\beta(8),\beta(10),\ldots,\beta(110) is irrational. These shorten the previously stated intervals with the same starting indices, whose upper endpoints are 9292 and 128128, respectively (twinheterodox, Hexagon 2611.00160v2, Section 8.2). The new bounds come from a different construction, based on powers of Krisman's hypergeometric kernel (Hexagon 2610.00285v2).

Author comments: 13 pages. Powers of a hypergeometric kernel give smaller shifted intervals of even Dirichlet beta values containing an irrational member. Includes exact seed and arithmetic certificates, source records, and an independent AI audit. Sources and checks accompany the submission.

Provenance statement

This work was developed with OpenAI Codex using GPT-6-Astra max. I directed the investigation, discussed the mathematical ideas, and reviewed and edited the exposition. Codex conducted literature searches, developed the technical arguments, drafted the manuscript, and produced the accompanying computational checks. My review covers the problem, the main result and parts of the reasoning; I have not independently verified the complete technical proof. A separate AI audit reported on 11 November 2026 found no substantive mathematical error or gap in the audited version and independently reconstructed the finite certificates. The auditor used fresh implementations after preliminary inspection and replay of the supplied checkers. The audit was not a blind review or formal verification, and the cited Lean project was not rebuilt. The current manuscript includes subsequent exposition and attribution revisions; the audit record identifies its input by hash. The manuscript, exact-arithmetic checkers, source records, audit report and independent audit implementations are available at https://github.com/twinheterodox/beta4/tree/main/powered-beta-kernels. The source ZIP includes the manuscript checkers, data and audit records under anc/.

Tools used

OpenAI
CodexVersion GPT-6-Astra max

References

  1. Shawn Krisman, One of β(4),β(6),…,β(36)\beta(4),\beta(6),\ldots,\beta(36) is irrational , 2026, Hexagon 2610.00285v2 (https://hexagonmath.org/pdf/2610.00285v2). Associated source: https://github.com/telaviv/beta-irrationality.link
  2. Li Lai and Johannes Sprang, Many pp-adic odd zeta values are irrational , version 2, 17 February 2025, arXiv:2306.10393v2 (https://arxiv.org/abs/2306.10393v2).arXiv
  3. Tanguy Rivoal, Irrationalité d'au moins un des neuf nombres ζ(5),ζ(7),…,ζ(21)\zeta(5),\zeta(7),\ldots,\zeta(21) , Acta Arith. 103 (2002), 157--167, doi:10.4064/aa103-2-5 (https://doi.org/10.4064/aa103-2-5).DOI
  4. V.\,N. Sorokin, On the Zudilin--Rivoal theorem , Math. Notes 81 (2007), 817--826, doi:10.1134/S000143460705029X (https://doi.org/10.1134/S000143460705029X).DOI
  5. twinheterodox, Irrational even beta values in shifted intervals , version 2, 6 November 2026, prepared with Codex, Hexagon 2611.00160v2 (https://hexagonmath.org/pdf/2611.00160v2). Associated source: https://github.com/twinheterodox/beta4.link
  6. Wadim Zudilin, Arithmetic of Catalan's constant and its relatives , Abh. Math. Semin. Univ. Hambg. 89 (2019), 45--53, doi:10.1007/s12188-019-00203-w (https://doi.org/10.1007/s12188-019-00203-w).DOI

Version history

  1. v1 · 13 pagesSubmitted by twinheterodox13 pages. Powers of a hypergeometric kernel give smaller shifted intervals of even Dirichlet beta values containing an irrational member. Includes exact seed and arithmetic certificates, source records, and an independent AI audit. Sources and checks accompany the submission.CurrentNov 11, 2026