For every fixed odd integer s≥5s\ge 5, the arithmetic nature of the Riemann zeta value ζ(s)=∑n≥1n−s\zeta(s)=\sum_{n\ge 1}n^{-s} is unknown: no proof of irrationality (or rationality) is known for any single such value, most famously ζ(5)\zeta(5). This question, recalled explicitly as open in the introduction of Laursen's 2025 paper on transcendence criteria for infinite products of algebraic numbers, is one of the oldest and most notorious problems in Diophantine approximation — the study of how closely real numbers can be approximated by rational numbers — and in the arithmetic of periods (numbers, such as π\pi and ζ(3)\zeta(3), that arise as integrals of algebraic functions over algebraic domains). All known results are collective — infinitely many odd zeta values are irrational (Rivoal), at least one of ζ(5),ζ(7),ζ(9),ζ(11)\zeta(5),\zeta(7),\zeta(9),\zeta(11) is irrational (Zudilin), at least two of ζ(5),…,ζ(35)\zeta(5),\dots,\zeta(35) are irrational (Lai–Zhou) — yet no individual odd value beyond ζ(3)\zeta(3) has been settled since Apéry's 1978 theorem.

Background

The Riemann zeta function is defined for real s>1s>1 by the convergent series ζ(s)=∑n=1∞n−s\zeta(s)=\sum_{n=1}^{\infty}n^{-s}. Euler's evaluation ζ(2)=π2/6\zeta(2)=\pi^2/6 and its generalization ζ(2n)∈π2n⋅Q\zeta(2n)\in\pi^{2n}\cdot\mathbb{Q} for positive integers nn, together with Lindemann's theorem that π\pi is transcendental (not a root of any nonzero polynomial with rational coefficients), settle the even values completely: every ζ(2n)\zeta(2n) is transcendental. The odd values are a different world. In 1978 Apéry stunned the community with a proof that ζ(3)\zeta(3) is irrational, constructed from explicit rational linear forms anζ(3)−bna_n\zeta(3)-b_n whose denominators grow slowly compared with how fast the forms tend to zero. Despite almost fifty years of intense effort, no odd value beyond ζ(3)\zeta(3) has yielded: for every fixed odd integer s≥5s\ge 5 it is not even known whether ζ(s)\zeta(s) is irrational, let alone transcendental.

All subsequent progress has been of a collective nature. Rivoal (2000), via the construction of explicit rational linear forms in 1,ζ(3),ζ(5),…1,\zeta(3),\zeta(5),\dots — developed in full with Keith Ball and published as Ball–Rivoal (2001) — proved that infinitely many of the numbers ζ(3),ζ(5),ζ(7),…\zeta(3),\zeta(5),\zeta(7),\dots are irrational. Zudilin (2002) refined the construction to show that at least one of the four numbers ζ(5),ζ(7),ζ(9),ζ(11)\zeta(5),\zeta(7),\zeta(9),\zeta(11) is irrational. Fischler–Sprang–Zudilin (2019) sharpened the quantitative lower bounds on how many odd zeta values must be irrational, Sprang (2018) gave a proof, by elementary means, that infinitely many odd zeta values are irrational, Lai (2024, 2025) improved the quantitative bounds further, and Lai–Zhou (2021) proved that at least two of ζ(5),ζ(7),…,ζ(35)\zeta(5),\zeta(7),\dots,\zeta(35) are irrational. The shared structural limitation of all these methods is that the produced linear forms involve several odd zeta values simultaneously, so the smallness-versus-denominator estimates certify only that some member of a finite set is irrational, never which one.

This state of affairs is what Laursen's 2025 paper on transcendence criteria for infinite products of algebraic numbers (an application of Schmidt's Subspace Theorem, a far-reaching generalization of the Thue–Siegel–Roth theorem bounding how closely an algebraic number can be approximated by rational numbers) recalls as motivation: we have a multitude of numbers known to be transcendental (ee, ζ(2n)\zeta(2n)), yet small perturbations of the series defining them — ∑n≥01/(n!+1)\sum_{n\ge 0}1/(n!+1) in place of e=∑n≥01/n!e=\sum_{n\ge 0}1/n!, or the odd zeta values in place of the even ones — render even the question of irrationality hopelessly open. Laursen's own results concern infinite products and do not touch the zeta question.

Problem Statement

For a fixed odd integer s≥5s\ge 5 — in particular for s=5s=5 — determine whether the value ζ(s)=∑n=1∞1ns\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}} is irrational. More generally, determine whether ζ(s)\zeta(s) is irrational for every odd integer s≥5s\ge 5. A determination of the irrationality of ζ(s)\zeta(s) for a single fixed odd integer s≥5s\ge 5 — in particular for the value s=5s=5 — resolves the question; the statement that ζ(s)\zeta(s) is irrational for every odd integer s≥5s\ge 5 is the strictly stronger general form of the same question, which a determination for a single fixed ss does not by itself settle. This is the question explicitly recalled as open in the introduction of Laursen (arXiv:2503.01575, 2025): the irrationality of ζ(s)\zeta(s) 'remains open when s≥5s\ge 5 is any fixed odd integer'. No answer, positive or negative, is currently known for any single such ss.

References

  1. Mathias L. Laursen, 'Transcendence criteria for infinite products of algebraic numbers', arXiv:2503.01575 (2025), https://arxiv.org/abs/2503.01575
  2. Tanguy Rivoal, 'La fonction Zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs', arXiv:math/0008051 (2000), https://arxiv.org/abs/math/0008051 (published in C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 267–270, doi:10.1016/S0764-4442(00)01624-4, https://doi.org/10.1016/S0764-4442(00)01624-4)
  3. Wadim Zudilin, 'Arithmetic of linear forms involving odd zeta values', arXiv:math/0206176 (2002), https://arxiv.org/abs/math/0206176 (published in J. Théorie Nombres Bordeaux 16 (2004), 251–291, doi:10.5802/jtnb.447, https://doi.org/10.5802/jtnb.447)
  4. Stéphane Fischler, 'Irrationalité de valeurs de zêta (d'après Apéry, Rivoal, ...)', Séminaire Bourbaki exposé 910, arXiv:math/0303066 (2003), https://arxiv.org/abs/math/0303066 (published in Astérisque 294 (2004), 27–62)
  5. Stéphane Fischler, Johannes Sprang, Wadim Zudilin, 'Many odd zeta values are irrational', arXiv:1803.08905 (2018), https://arxiv.org/abs/1803.08905 (published in Compositio Mathematica 155 (2019), 938–952, doi:10.1112/S0010437X1900722X, https://doi.org/10.1112/S0010437X1900722X)
  6. Johannes Sprang, 'Infinitely many odd zeta values are irrational. By elementary means', arXiv:1802.09410 (2018), https://arxiv.org/abs/1802.09410
  7. Li Lai, Li Zhou, 'At least two of ζ(5),ζ(7),…,ζ(35) are irrational', arXiv:2103.00904 (2021), https://arxiv.org/abs/2103.00904 (published in Publicationes Mathematicae Debrecen 101 (2022), 353–372, doi:10.5486/pmd.2022.9252, https://doi.org/10.5486/pmd.2022.9252)
  8. Li Lai, 'Small improvements on the Ball–Rivoal theorem and its p-adic variant', arXiv:2407.14236 (2024), https://arxiv.org/abs/2407.14236
  9. Li Lai, 'A note on the number of irrational odd zeta values, II', arXiv:2501.05321 (2025), https://arxiv.org/abs/2501.05321
  10. Keith Ball, Tanguy Rivoal, 'Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs', Inventiones Mathematicae 146 (2001), 193–207, doi:10.1007/s002220100168, https://doi.org/10.1007/s002220100168
  11. Li Lai, Johannes Sprang, Wadim Zudilin, 'A note on the irrationality of ζ₂(5)', arXiv:2505.05005 (2025), https://arxiv.org/abs/2505.05005 (concerns the 2-adic zeta value, not the classical ζ(5))
  12. Shekhar Suman, 'A note on the Irrationality of ζ(5) and Higher Odd Zeta Values', arXiv:2407.07121 (2024, revised through v7, May 2025), https://arxiv.org/abs/2407.07121 (unrefereed preprint; claim not accepted by the number-theory community)