The problem asks for the arithmetic nature of ζ(3)=∑n≥1n−3≈1.2020569\zeta(3)=\sum_{n\ge1}n^{-3}\approx1.2020569 (Apéry's constant): prove or disprove, unconditionally, that ζ(3)\zeta(3) is transcendental, and decide the stronger question of whether ζ(3)\zeta(3) and π\pi are algebraically independent over Q\mathbb{Q}. Apéry proved ζ(3)\zeta(3) irrational in 1979, and all subsequent progress has been quantitative irrationality bounds and Q\mathbb{Q}-linear independence among collections of odd zeta values; no odd zeta value is known to be transcendental or algebraically independent of π\pi. Gun–Murty–Rath (2014) proved unconditionally that at least one of '(ζ(3)/π3)2(\zeta(3)/\pi^3)^2 is irrational' and 'Γ3(1/2)/Γ2(1/2)\Gamma_3(1/2)/\Gamma_2(1/2) is transcendental' holds, and that Schanuel's conjecture implies at least one of 'ζ(3)\zeta(3) and π\pi are algebraically independent' and 'Γ3(1/2)/Γ2(1/2)\Gamma_3(1/2)/\Gamma_2(1/2) is transcendental' — so even the conditional picture is dichotomous. The unconditional transcendence of ζ(3)\zeta(3) remains completely open.

Background

The Riemann zeta function is defined for a complex variable ss with real part greater than 1 by ζ(s)=∑n=1∞n−s\zeta(s)=\sum_{n=1}^{\infty}n^{-s}, and extends analytically to all s≠1s\ne1. Its values behave very differently at even and odd positive integers: the even values are rational multiples of powers of π\pi (for example ζ(2)=π2/6\zeta(2)=\pi^2/6) and hence transcendental, since π\pi is transcendental; no analogous closed form is known for any odd value ζ(3),ζ(5),…\zeta(3),\zeta(5),\ldots. The number ζ(3)≈1.2020569\zeta(3)\approx1.2020569, called Apéry's constant, is one of the most basic 'periods' (numbers given by integrals of rational functions over domains defined by rational inequalities) and appears across number theory and physics. Apéry proved in 1979 that ζ(3)\zeta(3) is irrational, and it is still the only individual odd zeta value known to be irrational.

A complex number is algebraic if it is a root of a nonzero polynomial with rational coefficients, and transcendental otherwise. Numbers α1,…,αk\alpha_1,\ldots,\alpha_k are algebraically independent over Q\mathbb{Q} if no nonzero polynomial in kk variables with rational coefficients vanishes at (α1,…,αk)(\alpha_1,\ldots,\alpha_k). Since π\pi is transcendental, algebraic independence of ζ(3)\zeta(3) and π\pi is strictly stronger than transcendence of ζ(3)\zeta(3): if ζ(3)\zeta(3) were algebraic, the pair would be algebraically dependent.

Beyond irrationality, all unconditional results concern rational linear approximation and linear independence. An irrationality measure μ\mu of a number α\alpha is a bound of the form ∣α−p/q∣>1/qμ+ε|\alpha-p/q|>1/q^{\mu+\varepsilon} for all rationals p/qp/q with qq large; Rhin–Viola (2001) obtained the best current measure μ(ζ(3))<5.513891\mu(\zeta(3))<5.513891. Rivoal (2000), and Ball–Rivoal (2001), proved that infinitely many of the odd values ζ(3),ζ(5),ζ(7),…\zeta(3),\zeta(5),\zeta(7),\ldots are irrational — more precisely that the dimension of the Q\mathbb{Q}-vector space spanned by 1,ζ(3),ζ(5),…,ζ(s)1,\zeta(3),\zeta(5),\ldots,\zeta(s) grows at least logarithmically in ss — and 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. The strongest recent advance, due to Fischler (2026), improves the lower bound on the number of Q\mathbb{Q}-linearly independent elements among 1,ζ(3),…,ζ(s)1,\zeta(3),\ldots,\zeta(s) to at least 0.21s/log⁡s0.21\sqrt{s/\log s}, the first asymptotic improvement on Ball–Rivoal's bound. All of these are statements about linear independence over Q\mathbb{Q}; no algebraic-independence or transcendence statement is known for any odd zeta value, and even the irrationality of ζ(3)/π3\zeta(3)/\pi^3 is open.

The main conditional results come from Schanuel's conjecture, a central unproved conjecture of transcendence theory stating that if complex numbers x1,…,xnx_1,\ldots,x_n are linearly independent over Q\mathbb{Q}, then the field generated over Q\mathbb{Q} by x1,…,xn,ex1,…,exnx_1,\ldots,x_n,e^{x_1},\ldots,e^{x_n} has transcendence degree at least nn. Gun–Murty–Rath (2014) connected this to ζ(3)\zeta(3) through the Barnes multiple gamma functions Γm(x)\Gamma_m(x), the higher analogues of the classical gamma function obtained by iterating the functional equation Γm+1(x+1)=Γm+1(x) Γm(x+1)\Gamma_{m+1}(x+1)=\Gamma_{m+1}(x)\,\Gamma_m(x+1) with Γ1\Gamma_1 the ordinary gamma function and Γ2\Gamma_2 the Barnes GG-function; special values of these functions at rational points are built from derivatives of ζ\zeta at negative integers and logarithms of gamma values. They proved, unconditionally, that at least one of the following two statements holds: (ζ(3)/π3)2(\zeta(3)/\pi^3)^2 is irrational, or Γ3(1/2)/Γ2(1/2)\Gamma_3(1/2)/\Gamma_2(1/2) is transcendental; and, assuming Schanuel's conjecture, that at least one of the following holds: ζ(3)\zeta(3) and π\pi are algebraically independent, or Γ3(1/2)/Γ2(1/2)\Gamma_3(1/2)/\Gamma_2(1/2) is transcendental. Thus even under Schanuel's conjecture, current methods do not deduce the algebraic independence of ζ(3)\zeta(3) and π\pi — the conditional landscape itself is unresolved.

Problem Statement

Is Apéry's constant ζ(3)=∑n=1∞1n3≈1.2020569\zeta(3)=\sum_{n=1}^{\infty}\frac{1}{n^{3}}\approx1.2020569 transcendental? Prove, unconditionally, that ζ(3)\zeta(3) is transcendental over Q\mathbb{Q}, or prove that it is algebraic. More strongly, determine whether ζ(3)\zeta(3) and π\pi are algebraically independent over Q\mathbb{Q}; since π\pi is transcendental, an affirmative answer to the algebraic-independence question implies transcendence, and both questions are open in both directions — no unconditional result currently rules out ζ(3)\zeta(3) being algebraic, and none establishes its transcendence. The two are not independent — an affirmative answer to the algebraic-independence question settles transcendence as well, and a proof that ζ(3)\zeta(3) is algebraic settles the algebraic-independence question negatively — so those two outcomes each settle both questions at once. Conditional deductions (for example from Schanuel's conjecture) resolve only derived conditional questions.

References

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  2. Roger Apéry, Irrationalité de ζ(2) et ζ(3), Astérisque 61 (1979), 11–13, Zbl 0401.10049, open-access scan at https://geodesic.mathdoc.fr/item/AST_1979__61__11_0/
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