18 problems

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…

Kaneko's conjecture. The map

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…

Ihara–Kaneko–Zagier conjecture. All algebraic relations in the algebra $\mathcal{Z}$ of multiple zeta values are consequences of the extended double shuffle relations.

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 $\mathfrak{h}=\bigoplus_{k=-1}^{\infty}\mathfrak{h}_{k}$ 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…

Surjectivity conjecture. The map is an isomorphism:

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

Strong Broadhurst–Kreimer–Zagier conjecture.

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…

Let $\mathfrak{ls}$ be the Lie algebra of solutions to the linearized double shuffle relations, let $\mathsf{S}$ denote the space of even period polynomials, and let…

Prime-level weight-three conjecture.

Weight-two RDT generation conjecture. The RDT in follow from the seeded relations, the RDS in and, and the FDT in and. The source bases this on computation and does not state a pro…

Universal standard-relations conjecture. If $N=1$ or $2$, the map $\varphi_{\mathbb{R}}$ is injective, so the algebra of MPVs is isomorphic to $R_{RDS}$. If $N=p$ is prime with…

Binary-index conjecture. If $N=3$ or $N=4$, every MPV of level $N$ is a linear combination of MPVs

Weight-two RDT conjecture. All RDT in weight two are consequences of the RDS and FDT. The source reports numerical evidence through level $N=49$ for this completeness claim.

Distribution-at-ones conjecture. All distribution relations in with $x_j=1$ for every $j$ are consequences of regularized double shuffle relations (RDS) for MPVs of level $d$. This…

Extended double-shuffle conjecture. The induced map $\varphi_R$ is injective; equivalently, the algebra of Euler sums is isomorphic to $R_{\mathrm{EDS}}$.