Determining the $\mathbb{Q}$-dimension of the span of $1,\zeta(3),\zeta(5),\ldots,\zeta(s)$: linear independence of odd zeta values

For each odd integer $s\ge 3$, let $d(s)$ be the dimension of the $\mathbb{Q}$-vector space spanned by $1$ and the odd zeta values $\zeta(3),\zeta(5),\ldots,\zeta(s)$. The folklore conjecture that $\pi,\zeta(3),\zeta(5),\ldots$ are algebraically independent over $\mathbb{Q}$ predicts the maximal value $d(s)=(s+1)/2$. Unconditionally, after Apéry's proof that $\zeta(3)$ is irrational, the lower bound grew from Ball–Rivoal's $\bigl(1+o(1)\bigr)\log s/(1+\log 2)$ (2001) to Fischler's $0.21\sqrt{s/\log s}$ (J. London Math. Soc. 2026), while separate work bounds only the number of irrational values, most recently $1.284\sqrt{s/\log s}$ (Lai 2025). The exact dimension is unknown for every $s\ge 5$: even the irrationality of $\zeta(5)$ — and, stronger still, whether $d(5)\ge 3$ — is open. The problem remains open.

Background

For an integer $k\ge 2$, the Riemann zeta function is defined by $\zeta(k)=\sum_{n\ge 1}n^{-k}$. Euler's formula expresses every even value as a rational multiple of a power of $\pi$, so $\zeta(2),\zeta(4),\ldots$ are all transcendental; for the odd values $\zeta(3),\zeta(5),\ldots$ no comparable formula is known or expected. A folklore conjecture, recorded for instance by Fischler–Sprang–Zudilin (2019) and predicted by Grothendieck's period conjecture for mixed Tate motives (a broad conjectural framework describing exactly which algebraic relations can hold among the periods — special values of motivic integrals — of mixed Tate motives, a class of arithmetic motives to which zeta values belong), asserts that $\pi,\zeta(3),\zeta(5),\ldots$ are algebraically independent over the rationals $\mathbb{Q}$ (no nonzero polynomial with rational coefficients in finitely many variables vanishes when evaluated at them). In particular, the numbers $1,\zeta(3),\zeta(5),\ldots,\zeta(s)$ should be linearly independent over $\mathbb{Q}$ (no nontrivial rational linear combination of them should vanish), so their $\mathbb{Q}$-linear span, the set of all finite combinations $q_0+\sum_j q_j\zeta(j)$ with rational coefficients, should have the maximal dimension $d(s)=(s+1)/2$: one basis vector for $1$ and one for each of the $(s-1)/2$ odd zeta values. Determining $d(s)$ is the linear-independence form of the classical open problem on the arithmetic nature of odd zeta values.

All lower bounds on $d(s)$ rest on producing rational linear forms in $1$ and the odd zeta values that are simultaneously extremely small and have controlled integer denominators. Apéry (1978) built such forms for $\zeta(3)$ alone, proving $\zeta(3)\notin\mathbb{Q}$, hence $d(s)\ge 2$; to this day no other individual odd zeta value is known to be irrational. Rivoal (2000) proved that infinitely many odd zeta values are irrational, and Ball–Rivoal (2001), combining a well-poised hypergeometric series (a hypergeometric series whose parameters are arranged so that its partial fraction decomposition yields a rational linear combination of $1$ and odd zeta values) with Nesterenko's linear independence criterion (a general theorem converting a sequence of very small integer linear forms in given numbers into a proof that those numbers are linearly independent), obtained $d(s)\ge\frac{1-\varepsilon}{1+\log 2}\log s$ for every odd $s$ large enough in terms of $\varepsilon>0$ (logarithms natural). Fischler–Zudilin (2010) refined that criterion and proved that for some odd $i_1\le 139$ and $i_2\le 1961$ the four numbers $1,\zeta(3),\zeta(i_1),\zeta(i_2)$ are linearly independent. Constructions tuned to small ranges give finite records: at least one of $\zeta(5),\zeta(7),\zeta(9),\zeta(11)$ is irrational (Zudilin 2001), at least one of $\zeta(5),\ldots,\zeta(21)$ (Rivoal 2002, with an elementary proof up to $\zeta(25)$ by Zudilin 2018), and at least two of $\zeta(5),\ldots,\zeta(35)$ (Lai–Zhou 2022).

A parallel line bounds only how many of the values are irrational, without proving linear independence. Fischler–Sprang–Zudilin (2019) used linear forms whose coefficient structure is governed by the divisors of a primorial (the product of the first primes) to show that at least $2^{(1-\varepsilon)\log s/\log\log s}$ of $\zeta(3),\ldots,\zeta(s)$ are irrational; Lai–Yu (2020) improved this to $\bigl(1.192\ldots\bigr)\sqrt{s/\log s}$, and Lai (2025) to $1.284\,\sqrt{s/\log s}$. Most importantly for the present problem, Fischler (J. London Math. Soc., published 2026, preprint 2021) constructed non-explicit linear forms using Siegel's lemma (a classical pigeonhole-counting argument producing small nonzero integer solutions of underdetermined linear systems) and proved $d(s)\ge 0.21\sqrt{s}/\sqrt{\log s}$ for all sufficiently large odd $s$ — the first asymptotic improvement of the Ball–Rivoal dimension bound in two decades, yet still roughly a factor $\sqrt{s}$ below the conjectured $(s+1)/2$. On the other side, no nontrivial upper bound on $d(s)$ is known: no $\mathbb{Q}$-linear relation among distinct odd zeta values has ever been found, and none is expected.

Problem Statement

Determine, for every odd integer $s\ge 3$, the exact value

$$d(s)=\dim_{\mathbb{Q}}\operatorname{Span}_{\mathbb{Q}}\bigl(1,\zeta(3),\zeta(5),\ldots,\zeta(s)\bigr).$$

The conjectural answer is $d(s)=(s+1)/2$ for every odd $s\ge 3$; equivalently, the infinite sequence $1,\zeta(3),\zeta(5),\ldots$ is linearly independent over $\mathbb{Q}$, which would follow from the folklore conjecture that $\pi,\zeta(3),\zeta(5),\ldots$ are algebraically independent. A complete answer is a rigorous proof establishing matching lower and upper bounds for $d(s)$ for every odd $s\ge 3$; the trivial upper bound is $d(s)\le (s+1)/2$, and any proven upper bound strictly below it must rest on explicitly proven $\mathbb{Q}$-linear relations among the listed numbers. For orientation on the current frontier: $d(s)\ge 2$ for all $s\ge 3$ (Apéry), $d(s)\ge 0.21\sqrt{s}/\sqrt{\log s}$ for all sufficiently large odd $s$ (Fischler), and it is open whether $d(5)\ge 3$, i.e. whether $\zeta(5)\notin\operatorname{Span}_{\mathbb{Q}}(1,\zeta(3))$ — which would in particular settle the also open irrationality of $\zeta(5)$. Logarithms are natural.

References

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