The algebra of multiple zeta values (MZVs) is governed, conjecturally, by two linked statements: the Ihara–Kaneko–Zagier conjecture that the extended double shuffle relations generate all algebraic relations among MZVs (ker⁡ζ∗=IEDS\ker\zeta^*=I_{\mathrm{EDS}}), and Zagier's dimension conjecture that the Q\mathbb Q-space of weight-kk MZVs has the Fibonacci-type dimension dkd_k with generating function 1/(1−x2−x3)1/(1-x^2-x^3). Guo–Zhang (2010) reformulated the shuffle side of this framework using free Rota–Baxter algebras and explicitly left both conjectures open. Both conjectures remain fully open: the inclusion IEDS⊆ker⁡ζ∗I_{\mathrm{EDS}}\subseteq\ker\zeta^* is a theorem, the upper bound dim⁡QDk≤dk\dim_{\mathbb Q}D_k\le d_k follows from Brown's proof of Hoffman's conjecture, numerical and mod-2 verifications extend to weight 22, and work through 2026 continues on partial structures without claiming completeness or the dimension formula. A resolution would settle, among other consequences, the irrationality of all odd zeta values.

Background

Multiple zeta values (MZVs) are the real numbers ζ(s1,…,sk)=∑n1>⋯>nk≥11n1s1⋯nksk,\zeta(s_1,\ldots,s_k)=\sum_{n_1>\cdots>n_k\ge 1}\frac{1}{n_1^{s_1}\cdots n_k^{s_k}}, defined whenever s1≥2s_1\ge 2; the sum s1+⋯+sks_1+\cdots+s_k is called the weight and kk the depth. They generalize special values of the Riemann zeta function (the depth-one case ζ(s)\zeta(s)), go back to Euler — who found identities such as ζ(2,1)=ζ(3)\zeta(2,1)=\zeta(3) and ζ(4)=4ζ(3,1)\zeta(4)=4\zeta(3,1) — and are conjecturally the basic building blocks of the periods of mixed Tate motives over Z\mathbb Z (a class of motives assembled from the Tate objects Q(n)\mathbb Q(n), whose periods include logarithms); they also arise as constants in Feynman integrals in quantum field theory.

Two multiplicative structures act on MZVs. Nested summation gives the stuffle (or quasi-shuffle) product ∗*, for example ζ(a)ζ(b)=ζ(a,b)+ζ(b,a)+ζ(a+b)\zeta(a)\zeta(b)=\zeta(a,b)+\zeta(b,a)+\zeta(a+b); it makes the Q\mathbb Q-vector space H∗0\mathcal H^0_* spanned by bracket symbols [s1,…,sk][s_1,\ldots,s_k] with s1≥2s_1\ge 2 a commutative algebra, and the evaluation map ζ∗:H∗0→R\zeta^*:\mathcal H^0_*\to\mathbb R, [s1,…,sk]↦ζ(s1,…,sk)[s_1,\ldots,s_k]\mapsto\zeta(s_1,\ldots,s_k), is an algebra homomorphism. Representing MZVs as iterated integrals over words in two letters x0,x1x_0,x_1 gives a second product, the shuffle product ⊔⊔\sqcup\sqcup, on the same index set. Comparing coefficients of ∗* and ⊔⊔\sqcup\sqcup yields the (finite) double shuffle relations among MZVs.

Indices with s1=1s_1=1 diverge, and regularization is where the deepest relations live. Ihara–Kaneko–Zagier regularized the divergent indices (assigning them polynomials in an auxiliary variable TT, the shuffle and stuffle regularizations being related by an explicit map ρ\rho built from single zeta values) and thereby obtained the extended double shuffle relations: for every word w1w_1 with all indices ≥1\ge 1 (possibly divergent) and every convergent word w0w_0, a relation of the shape Z(reg∗(w1∗w0−w1⊔⊔w0))=0,Z\big(\mathrm{reg}_*(w_1 * w_0 - w_1 \sqcup\sqcup w_0)\big)=0, where reg∗\mathrm{reg}_* is the stuffle regularization map and ZZ the (regularized) evaluation. Let IEDSI_{\mathrm{EDS}} be the ideal of H∗0\mathcal H^0_* generated by these relations. Ihara–Kaneko–Zagier proved that IEDS⊆ker⁡ζ∗I_{\mathrm{EDS}}\subseteq\ker\zeta^*: all extended double shuffle relations are genuine identities among MZVs.

Guo–Zhang studied these structures through Rota–Baxter algebras. A Rota–Baxter algebra of weight 00 is a commutative algebra RR equipped with a linear operator P:R→RP:R\to R satisfying P(a)P(b)=P(aP(b)+bP(a))P(a)P(b)=P(aP(b)+bP(a)) — an abstraction of integration by parts for the ordinary integral operator. Guo–Zhang proved that the shuffle algebra of words in x0,x1x_0,x_1 (equivalently, the bracket algebra with the operator I([s1,s2,…,sk])=[s1+1,s2,…,sk]I([s_1,s_2,\ldots,s_k])=[s_1+1,s_2,\ldots,s_k]) is the free commutative nonunitary Rota–Baxter algebra on one generator, so that all shuffle relations for polylogarithms and regularized MZVs follow from the image of a single series, and that the regularization map ρ\rho of Ihara–Kaneko–Zagier arises naturally from renormalized diagonal MZVs (building on earlier work of Ebrahimi-Fard–Guo). This is a structural reformulation only: the paper itself states that it leaves the major arithmetic questions open.

Those questions are the subject of this problem. First, the completeness of extended double shuffle, conjectured by Ihara–Kaneko–Zagier: ker⁡ζ∗=IEDS\ker\zeta^*=I_{\mathrm{EDS}}, that is, every polynomial relation with rational coefficients among convergent MZVs is a consequence of extended double shuffle; the source paper notes that this would imply the irrationality of ζ(2n+1)\zeta(2n+1) for every n≥1n\ge 1. Second, Zagier's dimension conjecture (1994): the Q\mathbb Q-space DkD_k spanned by all MZVs of weight kk (products of MZVs of complementary weights are again MZVs by the stuffle product, so the MZV algebra is expected to be graded by weight) has dimension dim⁡QDk=dk,∑k≥0dkxk=11−x2−x3,\dim_{\mathbb Q}D_k=d_k,\qquad \sum_{k\ge 0}d_kx^k=\frac{1}{1-x^2-x^3}, i.e. d0=1d_0=1, d1=0d_1=0, d2=1d_2=1, dk=dk−2+dk−3d_k=d_{k-2}+d_{k-3} (so d22=200d_{22}=200); the source paper recalls the standard remark that this would imply algebraic independence of ζ(2)\zeta(2) and ζ(n)\zeta(n) for odd n≥3n\ge 3.

What is known: the motivic theory of MZVs (Terasoma; Deligne–Goncharov), culminating in Brown's proof of a conjecture of Hoffman, shows that every MZV is a Q\mathbb Q-linear combination of MZVs all of whose arguments are 22s and 33s; since there are exactly dkd_k such indices of weight kk, the upper bound dim⁡QDk≤dk\dim_{\mathbb Q}D_k\le d_k is proven. The reverse inequality is wide open — no technique can currently prove even dim⁡QD5=2\dim_{\mathbb Q}D_5=2 (the linear independence of the two conjectured basis elements of weight 5). Numerically, the multiple zeta value data mine provides proven reductions of all MZVs of weight at most 2222 to explicit bases whose dimensions agree with dkd_k, and computations of the mod-22 formal spaces cut out by binary extended double shuffle relations match the conjectured dimensions through weight 2222. The two conjectures are two facets of one objective: a complete description of the algebra of multiple zeta values — its presentation by generators and relations, and its graded dimensions.

Problem Statement

Resolve the following two linked conjectures, which together give a complete description of the algebra of multiple zeta values. Let H∗0\mathcal H^0_* denote the Q\mathbb Q-algebra of bracket symbols [s1,…,sk][s_1,\ldots,s_k] with s1≥2s_1\ge 2 under the quasi-shuffle (stuffle) product ∗*; let ζ∗:H∗0→R\zeta^*:\mathcal H^0_*\to\mathbb R be the evaluation homomorphism [s1,…,sk]↦ζ(s1,…,sk)[s_1,\ldots,s_k]\mapsto\zeta(s_1,\ldots,s_k); and let IEDS⊂H∗0I_{\mathrm{EDS}}\subset\mathcal H^0_* be the ideal generated by the extended double shuffle relations of Ihara–Kaneko–Zagier, i.e. by the elements reg∗(w1∗w0−w1⊔⊔w0)\mathrm{reg}_*(w_1*w_0-w_1\sqcup\sqcup w_0) for all words w1w_1 with indices ≥1\ge 1 and all convergent words w0w_0, where reg∗\mathrm{reg}_* is the stuffle regularization and ⊔⊔\sqcup\sqcup the shuffle product. It is known that IEDS⊆ker⁡ζ∗I_{\mathrm{EDS}}\subseteq\ker\zeta^*.

(Q1) Extended double shuffle completeness: is ker⁡ζ∗=IEDS\ker\zeta^*=I_{\mathrm{EDS}}? That is, is every polynomial relation with rational coefficients among convergent multiple zeta values a consequence of the extended double shuffle relations?

(Q2) Zagier's dimension conjecture: writing DkD_k for the Q\mathbb Q-space spanned by all MZVs of weight kk, is dim⁡QDk=dk\dim_{\mathbb Q}D_k=d_k for every kk, where ∑k≥0dkxk=1/(1−x2−x3)\sum_{k\ge 0}d_kx^k=1/(1-x^2-x^3) (equivalently d0=1d_0=1, d1=0d_1=0, d2=1d_2=1, dk=dk−2+dk−3d_k=d_{k-2}+d_{k-3})?

Either conjecture may be proved or refuted on its own; since both questions concern the same object — the quotient H∗0/ker⁡ζ∗\mathcal H^0_*/\ker\zeta^*, its presentation by generators and relations and its graded dimensions — a complete determination of one is expected to constrain the other. Partial results must retain the original generality: numerical verification in bounded weight, or a new proof of the known inclusion or of the known upper bound dim⁡QDk≤dk\dim_{\mathbb Q}D_k\le d_k, does not resolve the problem.

References

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