For even arguments, the Riemann zeta function takes values that are rational multiples of powers of π and hence transcendental, but for odd arguments almost nothing is known: since Apéry's 1978 announcement (published in 1979) only ζ(3) is known to be irrational. This problem records the open question that Fischler–Sprang–Zudilin explicitly note in "Many odd zeta values are irrational" (Compositio Math. 2019) — the irrationality of ζ(5) — together with the counting question that their theorem motivates: (1) prove that ζ(5) is irrational; and (2) determine the true order of the number N(s) of irrational values among ζ(3), ζ(5), …, ζ(s), which is conjectured to be maximal, N(s) = (s−1)/2, by the folklore conjecture (implied by Grothendieck's period conjecture) that π, ζ(3), ζ(5), … are algebraically independent; form (2) is a coherent quantitative counterpart of the same objective but is not itself posed as a question in that paper. The state of the art gives N(s) ≥ 1.284·√(s/log s) (Lai, building on Lai–Yu) and, after Fischler's 2026 theorem, the same order √(s/log s) for the dimension of the Q-vector space spanned by 1 and the odd zeta values up to s. Irrationality of any single odd zeta value beyond ζ(3), and any positive-proportion lower bound for N(s), remain open.

Background

The Riemann zeta function is defined for a complex argument s with Re(s) > 1 by ζ(s)=∑n≥1n−s\zeta(s)=\sum_{n\ge1}n^{-s}. Euler proved that at positive even integers the values are explicit rational multiples of powers of π: ζ(2)=π2/6\zeta(2)=\pi^2/6, ζ(4)=π4/90\zeta(4)=\pi^4/90, and generally ζ(2m)\zeta(2m) is a non-zero rational multiple of π2m\pi^{2m}; since π is transcendental (Lindemann, 1882), all even zeta values are transcendental. At odd integers no such structure exists, and the arithmetic nature is almost entirely unknown: a folklore conjecture, predicted by Grothendieck's period conjecture for mixed Tate motives (the statement that periods of these motives satisfy no algebraic relations beyond those forced structurally), asserts that the numbers π,ζ(3),ζ(5),ζ(7),…\pi,\zeta(3),\zeta(5),\zeta(7),\ldots are algebraically independent over the rationals (i.e. satisfy no non-zero polynomial relation with rational coefficients). In particular, conjecturally every odd zeta value is irrational, and among ζ(3),ζ(5),…,ζ(s)\zeta(3),\zeta(5),\ldots,\zeta(s) (odd ss) the number of irrational ones should be the maximum possible, (s−1)/2(s-1)/2.

The only individual odd zeta value known to be irrational is ζ(3)\zeta(3) (Apéry's theorem, announced in 1978 and published in 1979). All further progress is aggregate. Rivoal (2000) proved that infinitely many odd zeta values are irrational, and Ball–Rivoal (2001) proved the quantitative linear-independence bound: for any ε>0\varepsilon>0 and all odd ss sufficiently large in terms of ε\varepsilon,

dim⁡QSpanQ(1,ζ(3),ζ(5),…,ζ(s)) ≥ 1−ε1+log⁡2 log⁡s,\dim_{\mathbb{Q}}\mathrm{Span}_{\mathbb{Q}}\bigl(1,\zeta(3),\zeta(5),\ldots,\zeta(s)\bigr)\ \ge\ \frac{1-\varepsilon}{1+\log 2}\,\log s,

so that at least a constant times log⁡s\log s of these numbers are linearly independent over the rationals. The proofs rest on well-poised hypergeometric series — factorial-normalized series whose terms involve a specially parametrized rational function, and whose sums are rational linear combinations of 1 and odd zeta values — together with Nesterenko's linear independence criterion (1985), which deduces linear independence of numbers from the existence of integer linear combinations that are non-zero and simultaneously very small relative to the size of their coefficients. For small ranges, elimination arguments sharpen these statements without identifying which values are irrational: Zudilin (2001) proved that at least one of ζ(5),ζ(7),ζ(9),ζ(11)\zeta(5),\zeta(7),\zeta(9),\zeta(11) is irrational, and Rivoal–Zudilin (2018) that at least two among ζ(5),…,ζ(69)\zeta(5),\ldots,\zeta(69) are, improved to ζ(5),…,ζ(35)\zeta(5),\ldots,\zeta(35) by Lai–Zhou (2022).

Fischler–Sprang–Zudilin (2019) used twisted hypergeometric series (sums shifted by j/Dj/D for divisors dd of an auxiliary integer DD, expressed through the Hurwitz zeta function ζ(i,α)=∑n≥0(n+α)−i\zeta(i,\alpha)=\sum_{n\ge0}(n+\alpha)^{-i}) and an elimination step based on the positivity of generalized Vandermonde determinants det⁡[xjαi]\det[x_j^{\alpha_i}] to prove 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),\ldots,\zeta(s) are irrational — asymptotically more than any power of log⁡s\log s, but with no linear independence among them, in contrast to Ball–Rivoal. Their paper explicitly notes that the irrationality of ζ(5)\zeta(5) itself remains open. Subsequently the picture moved twice: Lai–Yu (2020), by an optimal placement of the zeros of the auxiliary rational functions, raised the count of irrational values to (c0−ε)s/log⁡s(c_0-\varepsilon)\sqrt{s/\log s} with c0=1.192507…c_0=1.192507\ldots, later improved to 1.284s/log⁡s1.284\sqrt{s/\log s} by Lai (2025) using the arithmetic factor Φn\Phi_n of Zudilin (2001); and Fischler (published 2026), constructing non-explicit linear forms via Siegel's lemma (the elementary statement that an underdetermined homogeneous linear system with integer coefficients has a non-zero integer solution of controlled height) combined with a refined Siegel-type criterion and a Shidlovsky-type multiplicity estimate (a tool bounding the possible rank degenerations of systems of functional approximations), proved that at least 0.21s/log⁡s0.21\sqrt{s/\log s} of the numbers 1,ζ(3),ζ(5),…,ζ(s)1,\zeta(3),\zeta(5),\ldots,\zeta(s) are linearly independent over the rationals — the first asymptotic improvement of the Ball–Rivoal dimension bound, which resolves the question of linear independence of a number of odd zeta values comparable to the known irrationality count. What remains untouched is the arithmetic of the individual values and the order of the counting function N(s)N(s) between the proved s/log⁡s\sqrt{s/\log s} and the conjectural (s−1)/2(s-1)/2.

Problem Statement

Let N(s)N(s) denote the number of irrational values among the odd zeta values ζ(3),ζ(5),…,ζ(s)\zeta(3),\zeta(5),\ldots,\zeta(s), for odd s≥3s\ge3. Form (1) below is the question that Fischler–Sprang–Zudilin (2019) explicitly record as open; form (2) is the counting question motivated by their theorem and by the folklore conjecture, and is not posed as such in that paper. The two linked forms share one objective, the arithmetic nature of the odd zeta values:

(1) Prove that ζ(5)\zeta(5) is irrational. Irrationality of a general individual value ζ(2n+1)\zeta(2n+1) with n≥2n\ge2 is the natural generalization of the same question, but ζ(5)\zeta(5) is the value named as open in the source and is the individual value targeted here. No odd zeta value other than ζ(3)\zeta(3) is currently known to be irrational, and all existing methods (well-poised hypergeometric series, twisted constructions, Siegel-lemma-based linear forms) prove only that some member of a finite family is irrational, never which one.

(2) Determine the true order of N(s)N(s) as s→∞s\to\infty through odd values: the best proved lower bounds are of order s/log⁡s\sqrt{s/\log s} (count: Lai–Yu, Lai; linear-independence dimension: Fischler), while the folklore conjecture that π,ζ(3),ζ(5),…\pi,\zeta(3),\zeta(5),\ldots are algebraically independent — a consequence of Grothendieck's period conjecture — predicts the maximum N(s)=(s−1)/2N(s)=(s-1)/2, i.e. that every odd zeta value is irrational. In particular, it is open whether a positive proportion of the odd zeta values are irrational, i.e. whether N(s)≥c⋅sN(s)\ge c\cdot s for some c>0c>0 and all large odd ss.

Both forms serve one objective, the arithmetic nature of the odd zeta values: for form (1), the specific value ζ(5)\zeta(5), where a proof of rationality would settle its nature just as much as a proof of irrationality and would refute the folklore conjecture; for form (2), the true order of N(s)N(s) as s→∞s\to\infty, conjecturally the maximum (s−1)/2(s-1)/2. A lower bound that improves on the current order s/log⁡s\sqrt{s/\log s} while leaving the true order of N(s)N(s) undetermined is progress on form (2) rather than a determination of that order. The conjectural endpoint — algebraic independence of π with all odd zeta values — is context for the conjectural value of N(s)N(s) and is not required for resolution.

References

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