40 problems

Kaneko's conjecture. The map

EDSR completeness conjecture. The extended double shuffle relations provide all relations among multiple zeta values; consequently, the duality relations should follow from EDSR.

Surjectivity conjecture. The map is an isomorphism:

Odd-coefficient convolution conjecture. For any integer $n\geq0$,

Racinet's conjecture. For $N=1$, there is a morphism of torsors

Let $N>1$, and consider multiple polylogarithms evaluated at $N$-th roots of unity. The double-shuffle and distribution relations are families of relations among these values. Prim…

Multiple zeta values are the periods defined by iterated integrals on the interval, and the regularized double shuffle relations are relations they satisfy. Regularized double shuf…

Depth-graded Ihara–Kaneko–Zagier conjecture. All algebraic relations in $\operatorname{gr}_D\mathcal{Z}$ are obtained by comparison of the maps

Let $\mathcal{Z}^f$ be the quotient of the shuffle algebra by the two-sided ideal generated by the formal extended double shuffle relations, and write $\zeta^f(k_1,\ldots,k_r)$ for…

Let $\mathcal{Z}$ be the algebra of multiple zeta values, and let the extended double shuffle relations be the relations obtained by combining the shuffle and series regularization…

Let $\mathrm{DMR}_0$ be Racinet's affine group scheme of solutions to the extended double shuffle relations, and let $U^{\rm dR}$ denote the prounipotent part of the motivic Galois…

Free odd generation conjecture. The double shuffle Lie algebra $\mathfrak{dm}$ is a free Lie algebra with exactly one generator in each odd weight $w\geq 3$:

Let $\mathcal{Z}$ be the $\mathbb{Q}$-algebra spanned by multiple zeta values … The extended double shuffle conjecture. All algebraic relations in $\mathcal{Z}$ are consequences of…

Multiple zeta values are defined, for positive integers $n_1,\ldots,n_r$ with $n_r\geq 2$, by … The double-shuffle conjecture asserts that, up to some renormalisation, all relation…

Multiple zeta values are real numbers given by convergent nested sums (with divergent cases handled by regularization), and a rational relation among them is a linear relation with…

For $m\geq 2$, let … be the double shuffle subspace, and let $\mathcal{M}_{k,m}$ be the naturally defined quotient of the dual of $\mathrm{Dsh}_m(k-m)$. Ihara–Kaneko–Zagier's dimen…

Let $\mathfrak{h}=\bigoplus_{k=-1}^{\infty}\mathfrak{h}_{k}$ be graded by … and write $\mathfrak{h}_{k}^{0}=\mathfrak{h}_{k}\cap\mathfrak{h}^{0}$. For $u,v,w\in\mathfrak{h}^{0}$, l…

Generation conjecture. The Lie algebra $\mathfrak{p}\mathfrak{ls}$ is generated by $\mathfrak{p}\mathfrak{ls}_1$.

Strong Broadhurst–Kreimer–Zagier conjecture.

Zagier's conjecture. All relations between multiple zeta values should be implied by double shuffle relations, suggesting

Let $\mathfrak{e}_m(\tau)$ be the mould of the elliptic multiple zeta values, and let the $\Delta$-bialternal relations be the elliptic double shuffle relations imposed on it. Dept…

Let $\mathfrak{e}_m(\tau)$ denote the mould of the elliptic multiple zeta values, and let the elliptic double shuffle relations be the $\Delta$-bialternality relations satisfied by…

Let multiple zeta values be the real numbers … for $d\in\mathbb{N}^{\ast}$ and $s_d,\ldots,s_1\in\mathbb{N}^{\ast}$ with $s_d\geq 2$. The double shuffle relations are the regulariz…

For each positive integer $w$, let $\FMZ_w$ be the $\Q$-vector space generated by finite multiple zeta values of weight $w$ and superbity one. Define $d_0=1$ and $d_w=\dim\FMZ_w$ f…

For weight $k$, let $eds_k$ denote the dimension of the span of the extended double-shuffle relations $ds(u,v)$ with $|u|+|v|=k$, $u\in\mathfrak H^0$, and…