HEXAGONbeta

One of β(4), β(6),..., β(36) is irrational

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

Contributed by Shawn Krisman

Version 2 / Oct 16, 2026 / CC BY 4.0

Abstract

Let β(s)=∑m≥0(−1)m(2m+1)−s\beta(s)=\sum_{m\ge 0}(-1)^m(2m+1)^{-s} be Dirichlet's beta function. We prove that at least one of the seventeen numbers β(4),β(6),…,β(36)\beta(4),\beta(6),\dots,\beta(36) is irrational. All previous explicit results of this kind that we know of include β(2)\beta(2), Catalan's constant; our statement does not involve β(2)\beta(2) at all. The proof uses the linear forms Ln=∑t≥0(−1)tRn′′(t)L_n=\sum_{t\ge0}(-1)^tR_n''(t) attached to a very-well-poised rational function RnR_n with triple integer zeros and half-integer poles. The second derivative, a β\beta-analogue of a device of Rivoal, removes β(2)\beta(2), and a symmetry removes the odd values. The zeros of RnR_n run through the region of its poles, which makes every prime p>np>n harmless. The common denominators are controlled by a pp-adic analysis and an arithmetic integral that we bound with an exact rational certificate. The size of LnL_n is bounded by an elementary estimate on the real axis, and its nonvanishing for infinitely many nn follows from a saddle-point analysis along a certified contour. The whole proof, including every numerical certificate, has been formalised in Lean 4 with Mathlib and checked by the Lean kernel, using only Lean's three standard axioms.

Provenance statement

This work was produced using two sessions of Opus 5.5. After OpenAI's proof of Catalan's Constant's (β(2)) irrationality we queried Claude to see what we can know about β(4)'s irrationality.

Tools used

Anthropic
Claude OpusVersion 5.5

References

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Version history

  1. v1Submitted by Shawn KrismanInitial depositSupersededOct 16, 2026
  2. v2Submitted by Shawn KrismanSubmitted as a new versionCurrentOct 16, 2026