For the Riemann zeta function, Euler's classical formula expresses every even value ζ(2n) as a rational multiple of a power of π, hence as a transcendental number; by contrast, after Apéry's 1978 proof that ζ(3) is irrational, no single odd value ζ(2n+1) with n ≥ 2 has ever had its irrationality established. Zudilin (2001) proved that at least one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational, and subsequent quantitative work has shown that many odd zeta values must be irrational, but every known result of this type is a non-constructive existence statement about a collection and does not identify any individual value. The problem is to determine the arithmetic nature — above all, irrationality — of an individually specified odd zeta value with 2n+1 ≥ 5, most famously ζ(5); in particular, to identify which member of {ζ(5), ζ(7), ζ(9), ζ(11)} is the irrational one guaranteed by Zudilin's theorem. The problem remains open as of 2026.

Background

The Riemann zeta function is defined for complex ss with Re(s)>1\mathrm{Re}(s)>1 by ζ(s)=∑k=1∞k−s\zeta(s)=\sum_{k=1}^{\infty}k^{-s} and extends analytically to all s≠1s\neq 1. Its values at positive even integers were determined by Euler: ζ(2n)\zeta(2n) is a rational multiple of π2n\pi^{2n} (for example ζ(2)=π2/6\zeta(2)=\pi^{2}/6), so every even value is transcendental, since π\pi is transcendental by the Lindemann–Weierstrass theorem. Nothing comparable is known at odd integers. In 1978–1979 Apéry stunned the field by proving that ζ(3)=∑k=1∞k−3\zeta(3)=\sum_{k=1}^{\infty}k^{-3} is irrational, via an explicit rapidly converging series and a continued fraction whose denominators grow slowly compared with the approximation quality; ζ(3)\zeta(3) is not even known to be transcendental, and for every odd integer 2n+1≥52n+1\ge 5 the most basic question — rational or irrational? — has remained unanswered since.

A standard route to irrationality is to construct explicit rational linear forms Lm=amθ+bmL_m=a_m\theta+b_m approximating a real number θ\theta so well that θ\theta cannot be rational: if am,bma_m,b_m are integers of controlled size and 0<∣Lm∣0<|L_m| tends to zero fast enough, θ\theta is irrational (one form of Nesterenko's linear-form criterion; hypergeometric integrals in the sense of Ball–Rivoal are the main tool for producing such forms for zeta values). Applying these ideas, Rivoal (2000) proved that infinitely many of the odd values ζ(3),ζ(5),ζ(7),…\zeta(3),\zeta(5),\zeta(7),\dots are irrational, and Ball and Rivoal (2001) showed that the dimension of the Q\mathbb{Q}-linear span of 1,ζ(3),ζ(5),…,ζ(s)1,\zeta(3),\zeta(5),\dots,\zeta(s) is at least (1−o(1)) log⁡s/(1+log⁡2)(1-o(1))\,\log s/(1+\log 2) as the odd integer s→∞s\to\infty. These are existence statements about collections: they force many irrational values to exist without exhibiting any specific one.

The strongest finite statement is due to Zudilin (2001): 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. Its key ingredient, the Φn\Phi_n (cyclotomic) factor method, exploits arithmetic factors related to Euler's totient function when estimating the denominators of the auxiliary linear forms. Zudilin later gave (2018) an elementary proof that one of ζ(5),ζ(7),…,ζ(25)\zeta(5),\zeta(7),\dots,\zeta(25) is irrational, and Lai and Zhou (2022) strengthened the finite statement to: at least two of ζ(5),ζ(7),…,ζ(35)\zeta(5),\zeta(7),\dots,\zeta(35) are irrational. On the asymptotic side, Fischler, Sprang and Zudilin (2019) introduced an elimination technique showing that at least 2(1−ε)log⁡s/log⁡log⁡s2^{(1-\varepsilon)\log s/\log\log s} of ζ(3),ζ(5),…,ζ(s)\zeta(3),\zeta(5),\dots,\zeta(s) are irrational; Lai and Yu (2020) improved this to at least (1.192507−ε)s/log⁡s(1.192507-\varepsilon)\sqrt{s/\log s} values, and Lai (2025) to at least 1.284s/log⁡s1.284\sqrt{s/\log s}. Very recently the same circle of ideas has been exported to pp-adic zeta values, analogues of the numbers ζ(s)\zeta(s) defined using a pp-adic notion of distance on the rationals (Lai–Sprang 2023–2025, Lai–Lupu–Sprang 2025). Yet none of these advances isolates an individual value: as Lai's 2025 preprint states plainly, the irrationality of ζ(5)\zeta(5) itself remains open, and no irrationality or transcendence statement is known for any single ζ(2n+1)\zeta(2n+1) with n≥2n\ge 2.

Problem Statement

Determine the arithmetic nature of an individually specified odd Riemann zeta value with 2n+1≥52n+1\ge 5: prove that a specific named value — most famously ζ(5)\zeta(5), or any single one of ζ(7),ζ(9),ζ(11),…\zeta(7),\zeta(9),\zeta(11),\dots — is irrational, or establish that it is rational. Whether such a value is transcendental or algebraic is a further, stronger question — open even for ζ(3)\zeta(3) — and deciding it is not required to resolve this problem. In particular, the celebrated special case is Zudilin's 2001 theorem, that at least one of the finite set {ζ(5),ζ(7),ζ(9),ζ(11)}\{\zeta(5),\zeta(7),\zeta(9),\zeta(11)\} is irrational: the theorem is non-constructive, and identifying which member is the irrational one is an instance of the general task. Statements that some unspecified members of a collection of odd zeta values are irrational (Rivoal-, Ball–Rivoal-, Zudilin-, or Lai–Yu-type results) do not name or settle any individual value.

References

  1. Roger Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), pp. 11–13; URL: https://numdam.org/item/AST_1979__61__11_0/
  2. Tanguy Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, Comptes Rendus de l'Académie des Sciences – Series I – Mathematics 331 (2000), no. 4, 267–270, DOI: 10.1016/S0764-4442(00)01624-4; URL: https://doi.org/10.1016/S0764-4442(00)01624-4
  3. Keith Ball and Tanguy Rivoal, Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs, Inventiones mathematicae 146 (2001), no. 1, 193–207, DOI: 10.1007/s002220100168; URL: https://doi.org/10.1007/s002220100168
  4. W. W. Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Russian Mathematical Surveys 56 (2001), no. 4, 774–776, DOI: 10.1070/RM2001v056n04ABEH000427; URL: https://doi.org/10.1070/RM2001v056n04ABEH000427
  5. Wadim Zudilin, One of the odd zeta values from ζ(5) to ζ(25) is irrational. By elementary means, Symmetry, Integrability and Geometry: Methods and Applications 14 (2018), article 028, DOI: 10.3842/SIGMA.2018.028; URL: https://doi.org/10.3842/SIGMA.2018.028
  6. Stéphane Fischler, Johannes Sprang and Wadim Zudilin, Many odd zeta values are irrational, Compositio Mathematica 155 (2019), no. 5, 938–952, DOI: 10.1112/s0010437x1900722x; URL: https://doi.org/10.1112/s0010437x1900722x
  7. Li Lai and Pin Yu, A note on the number of irrational odd zeta values, Compositio Mathematica 156 (2020), no. 8, 1699–1717, DOI: 10.1112/s0010437x20007307; URL: https://arxiv.org/abs/1911.08458
  8. Li Lai and Li Zhou, At least two of ζ(5), ζ(7), …, ζ(35) are irrational, Publicationes Mathematicae Debrecen 101 (2022), no. 3–4, 353–372, DOI: 10.5486/pmd.2022.9252; URL: https://doi.org/10.5486/pmd.2022.9252
  9. Li Lai, A note on the number of irrational odd zeta values, II, arXiv:2501.05321 (2025); URL: https://arxiv.org/abs/2501.05321
  10. Li Lai and Johannes Sprang, Many p-adic odd zeta values are irrational, arXiv:2306.10393 (2023, revised 2025), to appear in Michigan Mathematical Journal; URL: https://arxiv.org/abs/2306.10393
  11. Li Lai, Cezar Lupu and Johannes Sprang, On the irrationality of certain p-adic zeta values, arXiv:2505.23088 (2025); URL: https://arxiv.org/abs/2505.23088