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Irrationality measures for G±5π2/48G\pm5\pi^2/48 and L(2,χ−3)L(2,\chi_{-3})

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Sean Litsey ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

Building on OpenAI's determinant method for the irrationality of Catalan's constant G=∑j≥0(−1)j/(2j+1)2G=\sum_{j\ge0}(-1)^j/(2j+1)^2, we prove \[ \begin{gathered} \mu\Bigl(G-\frac{5\pi^2}{48}\Bigr)\le 38.35,\qquad \mu\Bigl(G+\frac{5\pi^2}{48}\Bigr)\le 38.35,\\ \mu\bigl(L(2,\chi_{-3})\bigr)\le 23.82 , \end{gathered} \] where μ\mu is the irrationality exponent. We know of no earlier irrationality measure for G±5π2/48G\pm5\pi^2/48. For L(2,χ−3)L(2,\chi_{-3}), which Calegari, Dimitrov and Tang proved irrational, the previous best bound we know of is their effective bound μ≤24781\mu\le24781. The approximations come from determinants of \emph{rectangular} L\"owner matrices of digamma differences at integer nodes. We bound their denominators with pp-adic assignment problems and rank deficits modulo pp. We prove nonvanishing modulo a prime by reducing it to bordered Krylov determinants, which we evaluate through Jacobi matrices and Darboux transformations. We bound their size with Andr\'eief's identity and a certified lower bound for a discrete logarithmic energy. Every result is formalized in Lean~4, with no unproved steps and only the standard axioms.

Provenance statement

This paper is a spinoff from the work hexagon:2610.00187. In the course of that work, we found constructions that, while not useful for more tightly bounding the irrationality exponent of Catalan's constant G, did apply to the three constants that are the subject of this paper. The paper is entirely due to AI. It was developed, proved, and formalized by Anthropic's Claude family of LLMs, running in Claude Code. The work was orchestrated by an Opus 5.5 instance running Opus 5.5, Sonnet 5.5, and Haiku 5.5 subagents, and was directed by the human author. The primary theorems are all formalized in Lean 4, available on GitHub.

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude SonnetVersion 5.5
Anthropic
Claude HaikuVersion 5.5
Anthropic
Claude Code
Lean
LeanVersion 4

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Version history

  1. v1Submitted by Sean LitseyInitial depositCurrentOct 20, 2026