For the Riemann zeta function ζ(s)=∑n=1∞n−s\zeta(s)=\sum_{n=1}^{\infty}n^{-s}, the values at even integers are classical rational multiples of powers of π\pi and hence transcendental, and ζ(3)\zeta(3) is irrational by Apéry's 1979 theorem. However, for every odd argument 2m+12m+1 with m≥2m\geq 2 — beginning with ζ(5)\zeta(5) — the arithmetic nature of ζ(2m+1)\zeta(2m+1) (rational, irrational, or transcendental) is unknown. Modern collective results show that infinitely many odd zeta values are irrational (Rivoal; Ball–Rivoal; Fischler–Sprang–Zudilin; Zudilin), and that at least one of ζ(5),ζ(7),ζ(9),ζ(11)\zeta(5),\zeta(7),\zeta(9),\zeta(11) is irrational, but no individual value beyond ζ(3)\zeta(3) has ever been identified as rational or irrational. The problem is to determine the arithmetic nature of an individual odd zeta value ζ(2m+1)\zeta(2m+1) with m≥2m\geq 2; the stronger universal form asks for a proof that ζ(2m+1)\zeta(2m+1) is irrational for every m≥1m\geq 1. Since the rationality of ζ(2m+1)\zeta(2m+1) implies the rationality of every power ζ(2m+1)k\zeta(2m+1)^{k}, and irrationality of ζ(2m+1)k\zeta(2m+1)^{k} implies irrationality of ζ(2m+1)\zeta(2m+1), progress on individual values bears directly on the arithmetic classification of powers of odd zeta values. The problem remains open.

Background

The Riemann zeta function is defined for complex ss with Re(s)>1\mathrm{Re}(s)>1 by the Dirichlet series ζ(s)=∑n=1∞n−s\zeta(s)=\sum_{n=1}^{\infty}n^{-s}. A real number is called irrational if it is not a ratio of integers, and transcendental if it is not a root of any nonzero polynomial with rational coefficients (transcendental numbers are in particular irrational). The arithmetic of zeta values — the question of which ζ(n)\zeta(n) are rational, irrational, or transcendental — goes back to the eighteenth century. At even arguments a classical closed formula expresses ζ(2m)\zeta(2m) as a rational multiple of π2m\pi^{2m}; since π\pi is transcendental, every even zeta value is transcendental. At odd arguments almost nothing unconditional is known about individual values. In 1979 Apéry stunned the community with a proof that ζ(3)\zeta(3) is irrational, based on an explicit sequence of rational approximations converging to ζ(3)\zeta(3) fast enough to force irrationality; this remains the only individual odd zeta value whose arithmetic nature is known. Subsequent progress has been collective rather than individual. Rivoal (2000) proved that infinitely many of the numbers ζ(3),ζ(5),ζ(7),…\zeta(3),\zeta(5),\zeta(7),\dots are irrational. Ball and Rivoal (2001) proved a quantitative version: the Q\mathbb{Q}-linear span (the vector space of rational linear combinations) of 1,ζ(3),ζ(5),…,ζ(s)1,\zeta(3),\zeta(5),\dots,\zeta(s) has dimension at least 1−ε1+log⁡2log⁡s\frac{1-\varepsilon}{1+\log 2}\log s for any ε>0\varepsilon>0 and ss large enough. Zudilin (2001) showed 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; later refinements (Zudilin 2018 by elementary means; Lai–Zhou 2022 proving at least two of ζ(5),…,ζ(35)\zeta(5),\dots,\zeta(35) irrational) sharpened the window, and Fischler, Sprang, and Zudilin (2019) proved that at least 2(1−ε)log⁡s/log⁡log⁡s2^{(1-\varepsilon)\log s/\log\log s} of the odd zeta values up to ζ(s)\zeta(s) are irrational. These results rest on constructing explicit families of linear forms (rational linear combinations of zeta values and 1) with controlled denominators and using Nesterenko-type linear independence criteria — techniques that bound how many values are simultaneously independent but fundamentally cannot pin down any single value. A closely related 2022 paper of Banerjee and Sahani derives Ramanujan-type transformation formulas for higher powers of odd zeta values (related to Lambert series, i.e. series of the shape ∑n≥1ane2πn−1\sum_{n\geq 1}\frac{a_n}{e^{2\pi n}-1}) and generalized Eisenstein series (modular-form analogues of such series); that paper explicitly notes the arithmetic nature of odd zeta values is unresolved, and its identities yield only disjunctive statements, e.g. that for mm odd at least one of ζ(2m+1)\zeta(2m+1) and ∑n≥11n2m+1(e2πn−1)\sum_{n\geq 1}\frac{1}{n^{2m+1}(e^{2\pi n}-1)} is transcendental, never an unconditional statement about ζ(2m+1)\zeta(2m+1) alone.

Problem Statement

Determine the arithmetic nature of the values of the Riemann zeta function at odd integers 2m+12m+1 with m≥2m\geq 2: prove that ζ(2m+1)\zeta(2m+1) is irrational — or rational — for at least one specific such value (starting with ζ(5)=∑n=1∞n−5\zeta(5)=\sum_{n=1}^{\infty}n^{-5}). A proof that ζ(2m+1)\zeta(2m+1) is irrational for every integer m≥1m\geq 1 — the universal strong form, of which only the case m=1m=1 (ζ(3)\zeta(3)) is settled — would be a strictly stronger result that in particular answers the single-value question, while an answer for one specific value leaves the arithmetic nature of every other odd zeta value open. Deciding the stronger question of whether such values are transcendental (not roots of any nonzero rational polynomial) would be a further strengthening, since irrationality alone is expected to be strictly weaker. Because the rationality of ζ(2m+1)\zeta(2m+1) implies the rationality of every power ζ(2m+1)k\zeta(2m+1)^{k}, a rationality answer for a given value settles the arithmetic nature of all powers of that same value.

References

  1. Soumyarup Banerjee, Vijay Sahani, Transformation formulas for the higher power of odd zeta values and generalized Eisenstein series, arXiv:2206.13331 (2022). https://arxiv.org/abs/2206.13331
  2. Roger Apéry, Irrationalité de ζ(2)\zeta(2) et ζ(3)\zeta(3), Astérisque 61 (1979), pp. 11–13. zbMATH Zbl 0401.10049. https://geodesic.mathdoc.fr/item/AST_1979__61__11_0/
  3. 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. https://doi.org/10.1016/S0764-4442(00)01624-4
  4. Keith Ball, 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. https://doi.org/10.1007/s002220100168
  5. Wadim Valentinovich Zudilin, One of the numbers ζ(5)\zeta(5), ζ(7)\zeta(7), ζ(9)\zeta(9), ζ(11)\zeta(11) is irrational, Uspekhi Matematicheskikh Nauk 56 (2001), no. 4, 149–150; English translation in Russian Mathematical Surveys 56 (2001), no. 4, 774–776. doi:10.4213/rm427. https://doi.org/10.4213/rm427
  6. Stéphane Fischler, Johannes Sprang, Wadim Zudilin, Many odd zeta values are irrational, Compositio Mathematica 155 (2019), no. 5, 938–952. arXiv:1803.08905. doi:10.1112/S0010437X1900722X. https://doi.org/10.1112/S0010437X1900722X
  7. Wadim Zudilin, One of the odd zeta values from ζ(5)\zeta(5) to ζ(25)\zeta(25) is irrational. By elementary means, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 14 (2018), paper 028. arXiv:1801.09895. doi:10.3842/SIGMA.2018.028. https://doi.org/10.3842/SIGMA.2018.028
  8. Li Lai, Li Zhou, At least two of ζ(5)\zeta(5), ζ(7)\zeta(7), …\ldots, ζ(35)\zeta(35) are irrational, Publicationes Mathematicae Debrecen 101 (2022), no. 3–4, 353–372. doi:10.5486/pmd.2022.9252. https://doi.org/10.5486/pmd.2022.9252
  9. Li Lai, Small improvements on the Ball–Rivoal theorem and its pp-adic variant, arXiv:2407.14236 (2024). https://arxiv.org/abs/2407.14236