HEXAGONbeta

At least one of β(4), β(6), …, β(32) is irrational

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(4),β(6),…,β(32)\beta(4),\beta(6),\ldots,\beta(32) is irrational. This reduces Krisman's seventeen-value set β(4),β(6),…,β(36)\beta(4),\beta(6),\ldots,\beta(36) (Hexagon 2610.00285v2) to fifteen values. The improvement comes from a sharper denominator estimate for his construction.

Author comments: 15 pages. An isolated-pole denominator refinement reduces Krisman's seventeen candidate even beta values to fifteen. Includes exact certificates, reproduction instructions and an independent mathematical audit with its verification scope.

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 computational checks. Two research sessions independently obtained the denominator refinement and fifteen-value conclusion from public sources, without access to each other's unpublished work; their arguments were subsequently compared and combined into this manuscript. My review covers the problem, the main result and parts of the reasoning; I have not independently verified the complete technical proof. A separate mathematical audit reported on 11 November 2026 found no mathematical error or gap affecting the theorem and independently reproduced the three computational inputs. Its report identifies the audited 14-page predecessor by hash. The current 15-page manuscript includes subsequent explanatory and layout revisions recorded separately. The audit accepts the stated classical inputs and is not formal verification or a priority assessment.

Tools used

OpenAI
CodexVersion GPT-6-Astra max

References

  1. S. 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/abs/2610.00285v2).hexagon
  2. L. Lai and J. Sprang, Many pp-adic odd zeta values are irrational , 2025, arXiv:2306.10393v2 (https://arxiv.org/abs/2306.10393v2).arXiv
  3. T. 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. twinheterodox, At least three of β(2),β(4),…,β(364)\beta(2),\beta(4),\ldots,\beta(364) are irrational , prepared with Codex, 2026, source, PDF, and certificates (https://github.com/twinheterodox/beta4/tree/94cf59dad9ebb2ff0ee666238a28abb14eca0466/finite-even-beta).link
  5. twinheterodox, Irrational even beta values from powered hypergeometric kernels , prepared with Codex, 2026, source, PDF, and certificates (https://github.com/twinheterodox/beta4/tree/94cf59dad9ebb2ff0ee666238a28abb14eca0466/powered-beta-kernels).link
  6. W. 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 · 15 pagesSubmitted by twinheterodox15 pages. An isolated-pole denominator refinement reduces Krisman's seventeen candidate even beta values to fifteen. Includes exact certificates, reproduction instructions and an independent mathematical audit with its verification scope.CurrentNov 11, 2026