HEXAGONbeta

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 1 / 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

References

  1. R. Apéry, Irrationalité de ζ(2)\zeta(2) et ζ(3)\zeta(3), Astérisque 61 (1979), 11--13, https://www.numdam.org/item/AST_1979__61__11_0/.link
  2. K. Ball and T. Rivoal, Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Invent.\ Math.\ 146 (2001), 193--207, https://doi.org/10.1007/s002220100168.DOI
  3. F. Calegari, Irrationality of certain pp-adic periods for small pp, Int.\ Math.\ Res.\ Not.\ 2005, no. 20, 1235--1249, https://doi.org/10.1155/IMRN.2005.1235; arXiv:math/0408214 (https://arxiv.org/abs/math/0408214).arXivDOI
  4. F. Beukers, Irrationality of some pp-adic LL-values, Acta Math.\ Sin.\ (Engl.\ Ser.) 24 (2008), 663--686, https://doi.org/10.1007/s10114-007-1029-2; arXiv:math/0603277 (https://arxiv.org/abs/math/0603277).arXivDOI
  5. F. Calegari, V. Dimitrov and Y. Tang, Arithmetic holonomy bounds and effective Diophantine approximation, in: Proceedings of the International Congress of Mathematicians 2026, Vol. 3, SIAM (2026), 356--376, https://doi.org/10.1137/25M1806776; arXiv:2510.04156 (https://arxiv.org/abs/2510.04156).arXivDOI
  6. F. Calegari, V. Dimitrov and Y. Tang, The linear independence of 11, ζ(2)\zeta(2), and L(2,χ−3)L(2,\chi_{-3}), preprint (2024), arXiv:2408.15403 (https://arxiv.org/abs/2408.15403).arXiv
  7. L. Lai, J. Sprang and W. Zudilin, A note on the irrationality of ζ2(5)\zeta_2(5), Int.\ Math.\ Res.\ Not.\ IMRN 2026, no. 16, rnag180, https://doi.org/10.1093/imrn/rnag180; arXiv:2505.05005 (https://arxiv.org/abs/2505.05005).arXivDOI
  8. L. Lai, On the irrationality of certain 22-adic zeta values, Int.\ J.\ Number Theory 21 (2025), 207--235, https://doi.org/10.1142/S1793042125500113; arXiv:2304.00816 (https://arxiv.org/abs/2304.00816).arXivDOI
  9. L. Lai and J. Sprang, Many pp-adic odd zeta values are irrational, Michigan Math.\ J., advance publication (2026), https://doi.org/10.1307/mmj/20236490; arXiv:2306.10393 (https://arxiv.org/abs/2306.10393).arXivDOI
  10. L. Lai, C. Lupu and J. Sprang, On the irrationality of certain pp-adic zeta values, Res.\ Math.\ Sci.\ 12 (2025), no. 4, Paper No. 77, https://doi.org/10.1007/s40687-025-00559-x; arXiv:2505.23088 (https://arxiv.org/abs/2505.23088).arXivDOI
Show all 35 referencesShow fewer references
  1. C. D. Long, Hybrid arithmetic holonomy and twenty-two individual pp-adic zeta values, preprint dated 24 August 2026, https://github.com/octonion/p-adic-zeta-irrationality, accessed 19 October 2026.link
  2. C. D. Long, Multiweight arithmetic holonomy and density-one irrationality of pp-adic zeta values, preprint dated 26 August 2026, https://github.com/octonion/p-adic-zeta-density, accessed 19 October 2026.link
  3. I. Kilani, ζ2(7)\zeta_2(7) is irrational via Hankel determinants, preprint dated 25 September 2026 and signed I. K., Zenodo, version 2 of 30 September 2026, https://doi.org/10.5281/zenodo.23058762 (all versions: https://doi.org/10.5281/zenodo.22961248).DOI
  4. S. Kintali, Irrationality of the 55-adic zeta value ζ5(3)\zeta_5(3), preprint dated 15 August 2026, https://shivakintali.github.io/papers/Zeta-5-3.pdf, accessed 4 October 2026.link
  5. T. Yi, A determinant argument for the irrationality of ζ7(3)\zeta_7(3). Proposed unconditional proof, preprint dated 15 September 2026, https://github.com/TaifengYi/zeta7-irrationality, accessed 4 October 2026.link
  6. I. Kilani, On the irrationality of ζ3(5)\zeta_3(5), preprint dated October 2026, Zenodo, 7 October 2026, https://doi.org/10.5281/zenodo.23203596; Lean 4 formalization at https://github.com/Ijtihed/zeta3-5-lean, accessed 19 October 2026.DOI
  7. A. Fauzan, ζ(5)\zeta(5) is irrational, preprint dated 17 September 2026, Zenodo, https://doi.org/10.5281/zenodo.22826419.DOI
  8. F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans.\ Comput.\ 66 (2017), 1281--1292, https://doi.org/10.1109/TC.2017.2690633; arXiv:1611.02831 (https://arxiv.org/abs/1611.02831).arXivDOI
  9. L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Grad.\ Texts in Math.\ 83, Springer, New York, 1997, https://doi.org/10.1007/978-1-4612-1934-7.DOI
  10. A. Volkenborn, Ein pp-adisches Integral und seine Anwendungen I, Manuscripta Math.\ 7 (1972), 341--373, https://doi.org/10.1007/BF01644073.DOI
  11. A. M. Robert, A Course in pp-adic Analysis, Grad.\ Texts in Math.\ 198, Springer, New York, 2000, https://doi.org/10.1007/978-1-4757-3254-2.DOI
  12. M. Bhargava, PP-orderings and polynomial functions on arbitrary subsets of Dedekind rings, J.\ Reine Angew.\ Math.\ 490 (1997), 101--127, https://doi.org/10.1515/crll.1997.490.101.DOI
  13. G. Szegő, Orthogonal Polynomials, 4th ed., Amer.\ Math.\ Soc.\ Colloq.\ Publ.\ 23, Amer.\ Math.\ Soc., Providence, RI, 1975, https://doi.org/10.1090/coll/023.DOI
  14. J. Hadamard, Sur la distribution des zéros de la fonction ζ(s)\zeta(s) et ses conséquences arithmétiques, Bull.\ Soc.\ Math.\ France 24 (1896), 199--220, https://doi.org/10.24033/bsmf.545.DOI
  15. Ch.-J. de la Vallée Poussin, Recherches analytiques sur la théorie des nombres premiers, Ann.\ Soc.\ Sci.\ Bruxelles 20 (1896), part 2, 183--256.
  16. J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J.\ Math.\ 6 (1962), 64--94, https://doi.org/10.1215/ijm/1255631807.DOI
  17. N. M. Katz, pp-adic properties of modular schemes and modular forms, in: Modular Functions of One Variable III (Antwerp, 1972), Lecture Notes in Math.\ 350, Springer, Berlin, 1973, 69--190, https://doi.org/10.1007/978-3-540-37802-0_3.DOI
  18. R. F. Coleman, pp-adic Banach spaces and families of modular forms, Invent.\ Math.\ 127 (1997), 417--479, https://doi.org/10.1007/s002220050127.DOI
  19. K. Buzzard, Analytic continuation of overconvergent eigenforms, J.\ Amer.\ Math.\ Soc.\ 16 (2003), 29--55, https://doi.org/10.1090/S0894-0347-02-00405-8.DOI
  20. F. Diamond and J. Shurman, A First Course in Modular Forms, Grad.\ Texts in Math.\ 228, Springer, New York, 2005, https://doi.org/10.1007/978-0-387-27226-9.DOI
  21. T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd ed., Grad.\ Texts in Math.\ 41, Springer, New York, 1990, https://doi.org/10.1007/978-1-4612-0999-7.DOI
  22. B. Conrad, Modular curves and rigid-analytic spaces, Pure Appl.\ Math.\ Q.\ 2 (2006), 29--110, https://doi.org/10.4310/PAMQ.2006.v2.n1.a3.DOI
  23. J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Grad.\ Texts in Math.\ 106, Springer, New York, 2009, https://doi.org/10.1007/978-0-387-09494-6.DOI
  24. R. Coleman and B. Mazur, The eigencurve, in: Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996), London Math.\ Soc.\ Lecture Note Ser.\ 254, Cambridge Univ.\ Press, Cambridge, 1998, 1--114, https://doi.org/10.1017/CBO9780511662010.003.DOI
  25. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Grundlehren Math.\ Wiss.\ 316, Springer, Berlin, 1997, https://doi.org/10.1007/978-3-662-03329-6.DOI

Version history

  1. v1Submitted by Antti HippeläinenInitial depositCurrentOct 20, 2026