HEXAGONbeta

Irrational even beta values in shifted intervals

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 positive odd integer mm, we prove that at least one of β(m+1),β(m+3),…,β(19m−1)\beta(m+1),\beta(m+3),\ldots,\beta(19m-1) is irrational, where β\beta is the Dirichlet beta function. More generally, for each positive integer BB we give an explicit constant CBC_B such that at least BB even beta values with indices between m+1m+1 and CBm−1C_Bm-1 are irrational, for every positive odd mm. Specific parameter choices give at least two irrational values among β(2),β(4),…,β(70)\beta(2),\beta(4),\ldots,\beta(70), at least three through β(186)\beta(186), and at least four through β(360)\beta(360).

Provenance statement

This work was developed through an extended interaction with OpenAI's Codex using Astra-6-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 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 in the underlying plain-text proof. The reviewer checked the general arguments for arbitrary allowed parameters and independently reproduced the finite calculations without running or importing the supplied checkers. The current PDF is a subsequent exposition of that argument. The added example from beta(4) through beta(198) was checked separately using the audited general criterion. The audit was not formal verification or certification of priority. The reviewer identity and model were not specified in the report. The manuscript, exact-arithmetic checkers, source records and audit reports are available at https://github.com/twinheterodox/beta4/tree/main/shifted-even-beta. The source ZIP includes the standalone checkers and their reference outputs under anc/.

Tools used

OpenAI
CodexVersion Astra-6-Max

References

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Version history

  1. v1 · 17 pagesSubmitted by twinheterodoxInitial depositSupersededNov 06, 2026
  2. v2 · 20 pagesSubmitted by twinheterodoxVersion 2 improves the uniform one-value endpoint from 67m-1 to 19m-1 and sharpens the general bound for any prescribed irrationality count. Specific parameter choices give at least two irrational values through beta(70), three through beta(186), and four through beta(360). The 20-page revision includes comparisons with v1, worked Gram and prime-dilation examples, and an independent audit of the underlying argument and exact certificates.CurrentNov 07, 2026