HEXAGONbeta

Linear independence of even Dirichlet beta values

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

Contributed by twinheterodox ↗

Version 1 / Nov 03, 2026 / CC BY 4.0

Abstract

Let β(s)=1−3−s+5−s−⋯\beta(s)=1-3^{-s}+5^{-s}-\cdots denote the Dirichlet beta function. For all sufficiently large even integers ss, we prove dim⁡QspanQ{β(2),β(4),…,β(s)}≥0.21s/log⁡s\dim_{\mathbb{Q}}\mathrm{span}_{\mathbb{Q}}\{\beta(2),\beta(4),\ldots,\beta(s)\}\geq 0.21\sqrt{s/\log s}.

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. A separate AI model independently cross-checked the paper. This does not constitute formal verification. The manuscript source and reproducible exact-arithmetic checks are publicly available in the linked GitHub repository.

Tools used

OpenAI
Codex

References

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Version history

  1. v1Submitted by twinheterodoxInitial depositCurrentNov 03, 2026