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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 2 / Oct 20, 2026 / CC BY 4.0

Abstract

We prove irrationality measures for several differences of values of the trigamma function ψ′\psi' at rational points. Among them are \[ \mu\bigl(L(2,\chi_{-3})\bigr)\le\MuChi,\qquad \mu\Bigl(L(2,\chi_{-3})\pm\frac{\pi^2}9\Bigr)\le\MuThirdsU, \] \[ \mu\Bigl(G\pm\frac{5\pi^2}{48}\Bigr)\le\MuLamU, \] and $\mu(\psi'(\frac k5)-\pi^2/6)\le\MuFifthsU$ and $\mu(\psi'(\frac k7)-\pi^2/6)\le\MuSeventhsU$ for kk prime to the denominator. For L(2,χ−3)L(2,\chi_{-3}) and L(2,χ−3)±π2/9L(2,\chi_{-3})\pm\pi^2/9 the best bound known before was 2478124781, due to Calegari, Dimitrov and Tang. For G±5π2/48G\pm5\pi^2/48, ψ′(k5)−π2/6\psi'(\frac k5)-\pi^2/6 and ψ′(k7)−π2/6\psi'(\frac k7)-\pi^2/6 we know of no earlier proof of irrationality. The approximations are determinants of rectangular L\"owner matrices built from the digamma function at integer nodes. We bound their denominators prime by prime, using the corank of the matrix modulo pp and cancellations at second and third order in its Smith normal form. Nonvanishing at rational points follows from Hermite interpolation in characteristic zero. The size of the determinants is bounded by Andr\'eief's identity and an energy estimate. The proof is formalized in Lean~4.

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 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, and GPT-6.1 Sol running in Codex. The work 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 depositSupersededOct 20, 2026
  2. v2Submitted by Sean LitseySubmitted as a new versionCurrentNov 03, 2026