HEXAGONbeta

Explicit lower bounds for the number of irrational even beta values

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

Contributed by twinheterodox ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

For every even integer s≥160s\ge160, we prove that at least ⌊π2/135 s/log⁡s⌋\left\lfloor\pi\sqrt{2/135}\,\sqrt{s/\log s}\right\rfloor of the numbers β(2),β(4),…,β(s)\beta(2),\beta(4),\ldots,\beta(s) are irrational, where β\beta is the Dirichlet beta function. We also give sharper finite endpoints for specified counts of irrational values

Provenance statement

This work was developed through an extended interaction with OpenAI's Codex. 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 understanding and review cover the problem, the main result, and parts of the reasoning, but I have not independently verified the complete technical proof. An independent mathematical audit, reported on 5 November 2026, found no substantive mathematical error or gap. The reviewer checked the general arguments and independently reproduced the finite calculations without running or importing the supplied checkers. The audit was not formal verification or a fresh audit of all cited literature. The manuscript, exact-arithmetic checkers, source records and audit reports are available at https://github.com/twinheterodox/beta4/tree/main/even-beta-counts. The source ZIP includes the standalone checkers and their reference outputs under anc/. Related paper: Linear independence of even Dirichlet beta values: https://hexagonmath.org/2611.00044v2

Tools used

OpenAI
CodexVersion Astra-6-Max

References

  1. V. Kumar and R. Tosi, Irrationality of values of LL-functions for real Dirichlet characters , preprint, 30 October 2026. arXiv:2610.41458v1 (https://arxiv.org/abs/2610.41458v1).arXiv
  2. L. Lai and J. Sprang, Many pp-adic odd zeta values are irrational , preprint, version of 17 February 2025. arXiv:2306.10393v2 (https://arxiv.org/abs/2306.10393v2).arXiv
  3. L. Lai and P. Yu, A note on the number of irrational odd zeta values , Compos. Math. 156 (2020), no. 8, 1699--1717. arXiv:1911.08458v2 (https://arxiv.org/abs/1911.08458v2).arXiv
  4. twinheterodox, Linear independence of even Dirichlet beta values , 3 November 2026. Hexagon 2611.00044v2 (https://hexagonmath.org/2611.00044v2).hexagon

Version history

  1. v1 · 20 pagesSubmitted by twinheterodoxInitial depositCurrentNov 05, 2026