Irrationality of $\zeta(5)$

Determine the arithmetic nature of the Riemann zeta value $\zeta(5)=\sum_{n\ge 1}n^{-5}$: prove that it is irrational, or prove that it is rational by exhibiting integers $p,q$ with $\zeta(5)=p/q$. This is the smallest odd zeta value left unsettled by Apéry's 1979 proof that $\zeta(3)$ is irrational. All known irrationality results for odd zeta values are collective — infinitely many are irrational (Rivoal 2000), at least one of $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational (Zudilin 2001), and at least $1.2845\,s^{1/2}/(\log s)^{1/2}$ of $\zeta(3),\zeta(5),\dots,\zeta(s)$ are irrational (Lai 2025, building on Fischler–Sprang–Zudilin 2019 and Lai–Yu 2020) — and none isolates any single value beyond $\zeta(3)$.

Background

The Riemann zeta function is defined for integers $s>1$ by $\zeta(s)=\sum_{n\ge 1}n^{-s}$; the values at odd integers $\zeta(3),\zeta(5),\zeta(7),\dots$ are called the odd zeta values, and their arithmetic nature is a central topic of transcendental number theory. Even values are completely understood — Euler expressed $\zeta(2k)$ as a rational multiple of $\pi^{2k}$, so all even zeta values are irrational (indeed transcendental) — but the odd values resist all classical tools. Apéry's 1979 theorem, that $\zeta(3)$ is irrational, remains the only individual odd zeta value settled.

Subsequent progress has been collective, establishing irrationality only for groups of values. Rivoal (2000) proved that infinitely many of the numbers $\zeta(3),\zeta(5),\zeta(7),\dots$ are irrational; Ball and Rivoal (2001) showed that the $\mathbb{Q}$-vector space spanned by $1$ and the odd zeta values up to $\zeta(s)$ has dimension growing at least logarithmically in $s$. Zudilin (2001) proved the sharpest 'small-degree' statement: at least one of $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational. In the asymptotic regime, Fischler–Sprang–Zudilin (2019) proved that the number of irrationals among $\zeta(3),\zeta(5),\dots,\zeta(s)$ is at least $2^{(1-\varepsilon)\log s/\log\log s}$ for every $\varepsilon>0$ and large enough $s$ (natural logarithms) — asymptotically more than any power of $\log s$; Lai and Yu (2020) improved this to order $s^{1/2}/(\log s)^{1/2}$ with explicit constant $1.1925\dots$, and Lai (2025) improved the constant to $1.2845\dots$. The conjectural truth, corresponding to the expectation that all odd zeta values are algebraically independent (in particular irrational), is a linear-in-$s$ count.

The methods behind all these results — hypergeometric ('Apéry-style') linear forms in zeta values, meaning explicit rational linear combinations $\sum a_k\,\zeta(k)$ that are simultaneously very small, ballast constructions, which adjoin auxiliary factors to control denominator growth, and the elimination technique of Fischler–Sprang–Zudilin — are inherently collective: they produce many candidate linear forms and conclude that some member of a group must vanish at a non-rational value, but cannot pinpoint which one. Consequently, the status of $\zeta(5)=1.0369277551\dots$, the smallest odd zeta value not covered by Apéry's theorem, has remained open since 1979. The closing discussion of Lai–Yu (2020) presents the individual irrationality of $\zeta(5)$ as unsettled, noting that except for Apéry's theorem the most remarkable result is Zudilin's theorem that at least one of $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational; its introduction likewise records how little is known about the conjectured algebraic independence of the odd zeta values.

Problem Statement

Determine the arithmetic nature of the fifth zeta value, the smallest odd zeta value not settled by Apéry's theorem:

$$\zeta(5)=\sum_{n=1}^{\infty}\frac{1}{n^{5}}=1.0369277551\dots$$

Either prove that $\zeta(5)$ is irrational, or prove that it is rational by exhibiting coprime integers $p,q$ with $q\ge 1$ such that $\zeta(5)=p/q$. The conjectured answer, implied by the standard expectation that the odd zeta values are algebraically independent over $\mathbb{Q}$, is irrationality. Apéry settled $\zeta(3)$ in 1979, and the irrationality of $\zeta(5)$ is presented as open in the closing discussion of Lai and Yu (2020).

References

1. Roger Apéry, Irrationalité de $\zeta(2)$ et $\zeta(3)$, Astérisque 61 (1979), pp. 11–13, NUMDAM identifier AST_1979__61__11_0, https://numdam.org/item/AST_1979__61__11_0/
2. T. Rivoal, La fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs, C. R. Acad. Sci. Paris 331 (2000), 267–270, DOI 10.1016/S0764-4442(00)01624-4, https://doi.org/10.1016/S0764-4442(00)01624-4
3. K. Ball and T. 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
4. W. Zudilin, One of the numbers $\zeta(5)$, $\zeta(7)$, $\zeta(9)$, $\zeta(11)$ is irrational, Russian Mathematical Surveys 56 (2001), 774–776, DOI 10.1070/RM2001v056n04ABEH000427, https://doi.org/10.1070/RM2001v056n04ABEH000427
5. S. Fischler, J. Sprang, W. Zudilin, Many odd zeta values are irrational, Compositio Mathematica 155 (2019), no. 5, 938–952, DOI 10.1112/S0010437X1900722X, https://doi.org/10.1112/S0010437X1900722X
6. Li Lai and Pin Yu, A note on the number of irrational odd zeta values, Compositio Mathematica 156 (2020), 1699–1717, DOI 10.1112/S0010437X20007307 (arXiv:1911.08458), https://doi.org/10.1112/S0010437X20007307
7. Li Lai and Li Zhou, At least two of $\zeta(5), \zeta(7), \ldots, \zeta(35)$ are irrational, Publications Mathematicae Debrecen 101 (2022), 353–372, DOI 10.5486/pmd.2022.9252, https://doi.org/10.5486/pmd.2022.9252
8. Stéphane Fischler, Linear independence of odd zeta values using Siegel's lemma, arXiv:2109.10136 (2021, revised 2025), https://arxiv.org/abs/2109.10136
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. Li Lai and Johannes Sprang, Many $p$-adic odd zeta values are irrational, Michigan Mathematical Journal (2026), DOI 10.1307/mmj/20236490, https://doi.org/10.1307/mmj/20236490