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On the irrationality of the 33-adic zeta value ζ3(5)\zeta_3(5)

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

Contributed by Antti Hippeläinen

Version 2 / Oct 20, 2026 / CC BY 4.0

Abstract

We give a proof that the 33-adic zeta value ζ3(5)\zeta_3(5) is irrational. The proof adapts to p=3p=3 a Hankel-determinant method that a recent preprint applies to ζ2(7)\zeta_2(7). A Volkenborn integral of an explicit rational function defines a linear functional with values in Q+Q ζ3(5)\mathbb{Q}+\mathbb{Q}\,\zeta_3(5), and its Hankel determinants are the values at ζ3(5)\zeta_3(5) of polynomials with rational coefficients. We bound the 33-adic valuation of these values and the size and the denominators of the coefficients. The new ingredients at p=3p=3 are a triangular basis of the polynomials that are integer valued on 3−1Z3×3^{-1}\mathbb{Z}_3^\times, a local estimate at primes of the order of nn in which a pole of order 33 coming from harmonic sums cancels exactly against a zero of order 33 of the rational function, and a non-vanishing theorem at an auxiliary prime, in which Wilson's theorem makes one block of a reduced Gram matrix invertible. Here nn is the parameter of the rational function. Two constants of the estimates are bounded by finite computations, and with these bounds the 33-adic valuation exceeds the size and the denominators together by more than 4.54.5 bits per n2n^2. An appendix gives a second proof, independent of the first, which adapts to ζ3(5)\zeta_3(5) and the modular curve X0(3)X_0(3) a determinant argument that a preprint of Yi gives for ζ7(3)\zeta_7(3), and which rests on published theorems on overconvergent modular forms.

Provenance statement

The note was produced by large language models in September and October 2026, in particular Anthropic's Claude Opus 5.5 running as autonomous agents. They adapted to p = 3 the Hankel-determinant method of the preprint of I. Kilani on the irrationality of ζ_2(7) and, for the second proof, the determinant argument of the preprint of T. Yi on ζ_7(3). They found and wrote the new steps, wrote and ran the programs, and wrote the text. The arguments were checked only by other instances of the same models, acting as referees and recomputing the numbers with programs of their. No human expert has checked either proof. The submitting contributor has read the note and checked its references. The programs for the computations are openly available at https://github.com/ahippelainen/zeta-3-5-irrational-public.

Tools used

Anthropic
Claude OpusVersion 5.5
Python Software Foundation and others
Python 3.12, python-flint 0.9.0, SymPy 1.14.0, NumPy 2.5.4

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Version history

  1. v1Submitted by Antti HippeläinenInitial depositSupersededOct 20, 2026
  2. v2Submitted by Antti HippeläinenSubmitted as a new versionCurrentOct 28, 2026