Euler's formula expresses every even Riemann zeta value ζ(2n)\zeta(2n) as a rational multiple of π2n\pi^{2n}, so all even values are transcendental; for odd arguments no analogous closed form is known — "finding a closed form for ζ(2n+1)\zeta(2n+1) has remained an open problem", as the cited source (Chamberland–Lopatto 2024) records. The problem is to find a closed form for the odd zeta values: decide whether, for some integer n≥1n\ge1, ζ(2n+1)\zeta(2n+1) is a rational multiple of π2n+1\pi^{2n+1} — that is, whether ζ(2n+1)=q π2n+1\zeta(2n+1)=q\,\pi^{2n+1} for some q∈Qq\in\mathbb{Q} — exhibiting such an identity or proving that none exists. The best exact formulas known are the Plouffe-type rapidly convergent series relating ζ(2n+1)\zeta(2n+1) and π2n+1\pi^{2n+1} to three explicit infinite series, conjectured by Plouffe and proved for all nn by Chamberland and Lopatto; they provide efficient computation but no closed form, and the source remarks that "it does not seem that ζ(2n+1)\zeta(2n+1) is a rational multiple of π2n+1\pi^{2n+1}".

Background

The Riemann zeta function is defined for a complex variable ss by ζ(s)=∑k≥1k−s\zeta(s)=\sum_{k\ge1}k^{-s}, converging for Re(s)>1\mathrm{Re}(s)>1; its values at positive integers are among the most classical constants of analysis. Euler (1735) discovered the closed form for even arguments: ζ(2n)\zeta(2n) equals a rational multiple of π2n\pi^{2n} (the rational factor is (−1)n+1B2n(2π)2n/(2(2n)!)(-1)^{n+1}B_{2n}(2\pi)^{2n}/(2(2n)!) in terms of the Bernoulli numbers B2nB_{2n}, a standard sequence of rational numbers generated by t/(et−1)=∑Bmtm/m!t/(e^t-1)=\sum B_m t^m/m!). Since Lindemann proved in 1882 that π\pi is transcendental (not a root of any nonzero polynomial with rational coefficients), every π2n\pi^{2n}, and hence every ζ(2n)\zeta(2n), is transcendental — in particular irrational. For odd arguments no analogous formula exists, and the source describes the state of the search: "The hunt for a clean result like Euler's has largely been abandoned, leaving researchers with the goal of finding formulas which either converge quickly or have an elegant form." On the arithmetic side, irrationality means not expressible as a ratio of two integers, and in 1979 Apéry proved precisely that much for the first odd value, showing ζ(3)=∑k≥1k−3\zeta(3)=\sum_{k\ge1}k^{-3} is irrational by constructing explicit sequences of rationals am/bma_m/b_m with ∣ζ(3)−am/bm∣|\zeta(3)-a_m/b_m| shrinking faster than 1/bm1+δ1/b_m^{1+\delta} (a landmark documented in van der Poorten's report "A Proof that Euler Missed"). No odd value beyond ζ(3)\zeta(3) has had its arithmetic nature settled since, and no odd value has ever received a closed form.

The current best exact representations of the odd values are the Plouffe-type series. Letting Sn(r)=∑k≥11/(kn(eπrk−1))S_n(r)=\sum_{k\ge1}1/(k^n(e^{\pi rk}-1)), Simon Plouffe conjectured identities relating π2n+1\pi^{2n+1} and ζ(2n+1)\zeta(2n+1) to three such series — for instance ζ(3)=28S3(1)−37S3(2)+7S3(4)\zeta(3)=28S_3(1)-37S_3(2)+7S_3(4) and ζ(5)=24S5(1)−25910S5(2)−110S5(4)\zeta(5)=24S_5(1)-\tfrac{259}{10}S_5(2)-\tfrac{1}{10}S_5(4) — discovered by integer-relation (PSLQ) searches. Their engine is a formula from Ramanujan's notebooks (Entry 21(i), Chapter 14 of Berndt's edition), and Chamberland and Lopatto (2024) found the general pattern of rational coefficients and proved the resulting identities for all nonnegative integers nn. That paper frames the remaining open question exactly: "While it does not seem that ζ(2n+1)\zeta(2n+1) is a rational multiple of π2n+1\pi^{2n+1}, a result in Ramanujan's notebooks gives a relationship with rapidly convergent infinite series." (Since π\pi is transcendental, π2n+1\pi^{2n+1} is transcendental, and any nonzero rational multiple q π2n+1q\,\pi^{2n+1}, q∈Qq\in\mathbb{Q}, is transcendental and hence irrational: a positive closed-form answer would immediately settle the arithmetic nature of that value. The converse fails — a nonzero rational multiple of π2n+1\pi^{2n+1} is itself irrational — so irrationality results, however strong, leave the closed-form question untouched.)

A separate line of research addresses the arithmetic nature of the odd zeta values. Modern attacks use Hermite–Padé approximation — simultaneous rational approximation to several related functions, producing small integral linear forms in 11 and several zeta values — combined with linear independence criteria such as Nesterenko's (which conclude irrationality or linear independence from good simultaneous Diophantine approximation). These techniques are inherently distributional: Ball and Rivoal (2001) proved that infinitely many of ζ(3),ζ(5),ζ(7),…\zeta(3),\zeta(5),\zeta(7),\dots are irrational by bounding from below the dimension of the Q\mathbb{Q}-vector space spanned by 11 and the odd values, and Zudilin (2001) sharpened the localization to: at least one of ζ(5),ζ(7),ζ(9),ζ(11)\zeta(5),\zeta(7),\zeta(9),\zeta(11) is irrational. But a dimension bound on a span can never single out one fixed index, and none of these results yields a closed-form identity or decides whether any ζ(2n+1)\zeta(2n+1) is a rational multiple of π2n+1\pi^{2n+1}. Quantitative refinements of the counting bounds and pp-adic analogues continue to appear (Fischler–Sprang–Zudilin 2019; Lai–Yu 2020; Lai 2024; Lai–Sprang 2026) without changing this picture.

Problem Statement

Find a closed form for the Riemann zeta function's odd values ζ(2n+1)=∑k≥1k−(2n+1)\zeta(2n+1)=\sum_{k\ge1}k^{-(2n+1)}, n≥1n\ge1, analogous to Euler's formula for the even values: decide whether ζ(2n+1)\zeta(2n+1) is a rational multiple of π2n+1\pi^{2n+1} — that is, whether there exist an integer n≥1n\ge1 and a rational qq with ζ(2n+1)=q π2n+1\zeta(2n+1)=q\,\pi^{2n+1}, exhibiting such an identity with proof, or proving that no such identity exists. This is the open problem recorded by the cited source (Chamberland–Lopatto 2024): "finding a closed form for ζ(2n+1)\zeta(2n+1) has remained an open problem", and "it does not seem that ζ(2n+1)\zeta(2n+1) is a rational multiple of π2n+1\pi^{2n+1}". At present no closed form is known for any odd argument: the strongest exact results are the Plouffe-type rapidly convergent series, now proved for all nn, which express ζ(2n+1)\zeta(2n+1) in terms of π2n+1\pi^{2n+1} and residual series Sn(r)S_n(r) that themselves have no known closed evaluation; on the arithmetic side, irrationality is known only for ζ(3)\zeta(3) (Apéry 1979) and in distributional form (infinitely many odd values are irrational; at least one of ζ(5),ζ(7),ζ(9),ζ(11)\zeta(5),\zeta(7),\zeta(9),\zeta(11) is), which by the remark above cannot settle the closed-form question in either direction.

References

  1. Marc Chamberland and Patrick Lopatto, "Formulas for odd zeta values and powers of π\pi", 2024, arXiv:2406.00567, DOI 10.48550/arXiv.2406.00567, https://arxiv.org/abs/2406.00567
  2. Keith Ball and Tanguy Rivoal, "Irrationalité d'une infinité de valeurs de la fonction zêta aux entiers impairs", Inventiones mathematicae 146, no. 1 (2001), 193–207, DOI 10.1007/s002220100168, https://doi.org/10.1007/s002220100168
  3. Tanguy Rivoal, "Irrationalité d'au moins un des neuf nombres zeta(5), zeta(7),...,zeta(21)", Acta Arithmetica 103, no. 2 (2002), 157–167, DOI 10.4064/aa103-2-5, https://doi.org/10.4064/aa103-2-5 (preprint arXiv:math/0104221, 2001, https://arxiv.org/abs/math/0104221)
  4. Wadim Zudilin, "Irrationality of values of the Riemann zeta function", Izvestiya: Mathematics 66, no. 3 (2002), 489–542, DOI 10.1070/IM2002v066n03ABEH000387, https://doi.org/10.1070/IM2002v066n03ABEH000387 (preprint arXiv:math/0104249, 2001, titled "Irrationality of values of zeta-function", https://arxiv.org/abs/math/0104249)
  5. Wadim 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, no. 4 (2001), 149–150, DOI 10.4213/rm427, https://doi.org/10.4213/rm427
  6. Stéphane Fischler, Johannes Sprang, and Wadim Zudilin, "Many odd zeta values are irrational", Compositio Mathematica 155, no. 5 (2019), 938–952, DOI 10.1112/S0010437X1900722X, 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, no. 8 (2020), 1699–1717, DOI 10.1112/S0010437X20007307, https://doi.org/10.1112/S0010437X20007307 (preprint arXiv:1911.08458, https://arxiv.org/abs/1911.08458)
  8. Li Lai, "Small improvements on the Ball–Rivoal theorem and its pp-adic variant", 2024, arXiv:2407.14236, DOI 10.48550/arXiv.2407.14236, https://arxiv.org/abs/2407.14236
  9. Li Lai and Johannes Sprang, "Many pp-adic odd zeta values are irrational", Michigan Mathematical Journal (2026, advance publication), DOI 10.1307/mmj/20236490, https://doi.org/10.1307/mmj/20236490
  10. Alfred van der Poorten, "A Proof that Euler Missed...", reprint in Pi: A Source Book, Springer (1997), DOI 10.1007/978-1-4757-2736-4_49 (originally Mathematical Intelligencer 1 (1979), documenting Apéry's proof that ζ(3)\zeta(3) is irrational), https://doi.org/10.1007/978-1-4757-2736-4_49