243 problems

Let $Z(3)$ denote the finite multiple-zeta value corresponding to $\zeta(3)$ under the Kaneko–Zagier correspondence, defined from Bernoulli numbers, and let $Q$ denote the quotient…

Todd's relation conjecture asserts that all $\mathbb{F}_q(\theta)$-linear relations among Thakur's multiple zeta values in positive characteristic are generated from the fundamenta…

Let $\mathcal{A}$ be the set of truly alternating multiple mixed values, and call $x\in\mathcal{A}$ unramified if it can be expressed as a $\mathbb{Q}$-linear combination of multip…

Let $w$ be a positive integer. An alternating multiple zeta value is a value of the alternating multiple zeta function of weight $w$. For a vector $\vec{s}$, write $w(\vec{s})$ for…

Let $\zeta(s_1,\ldots,s_r)$ be a multiple zeta value with $s_1>1$ and $s_2,\ldots,s_r\ge1$, and let its weight be $s_1+\cdots+s_r$. For $k\ge1$, define … and…

Hilbert–Poincaré rationality conjecture. The Hilbert–Poincaré series of these coefficient spaces is rational:

Kaneko–Zagier conjecture. There exists an isomorphism

Let $f\colon \mathbb{N}\rightarrow [1,+\infty)$, and let $B_f$ be the set of real numbers $\alpha\in [1,+\infty)$ such that … for infinitely many $r,k_1,\ldots,k_r$ with…

Hoffman basis conjecture. The Hoffman elements $\zeta(k_1,\ldots,k_d)$ with $k_i\in\{2,3\}$ form a basis for $\mathcal{Z}$; in particular,

Zero-measure conjecture.

Broadhurst–Kreimer's conjecture. The generating series of the dimensions of the weight- and depth-graded parts is

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

For $N=2$ or $4$, even integers $k\ge4$, and integers $j$ with $1\le j\le (k-2)/2$, define coefficients $a_i$ and $b_i$ by … and … Here $P_{N,k,j}^{(+)}$ and $P_{N,k,j}^{(-)}$ are…

Let $\mathsf{ES}_w$ be the $\mathbb{Q}$-vector space generated by Euler sums of weight $w$, and let $F_0=F_1=1$ and $F_k=F_{k-1}+F_{k-2}$ for $k\geq2$. Zlobin's Euler sum basis con…

Uneven Broadhurst–Kreimer conjecture. The bigraded dimensions of the totally odd part are given by

Let $\mathcal{P}_{\phi^3}^{\mathfrak{m}}$ denote the space of motivic periods of six-dimensional $\phi^3$ theory, and let $\mathcal{P}^{\mathfrak{dr}}$ denote the de Rham period sp…

Todd's dimension conjecture. For every positive integer $w$,

Central binomial sum conjecture.

Let $S_{N,r}$ be the index set used to define the coefficient space $\mathbb{P}_{N,r}$, let $\mathbf{W}_{N,r}\subseteq\mathbb{P}_{N,r}$ be the space of polynomials satisfying the s…

Let $\mathfrak{dg}_1$ be the depth-one component of the depth-graded motivic Lie algebra, let $\mathbb{P}$ be the space of restricted even period polynomials, and let…

Tasaka's isomorphism conjecture. The map $\widetilde{\xi}$ is an isomorphism. The conjecture concerns the depth-three case and is attributed to Tasaka. The paper does not report a…

Let $I_n$ denote the coefficients in Louchard's asymptotic expansion, and let $\zeta(i)$ be the ordinary Riemann zeta values. Hoffman's rational zeta-value conjecture. For each $n$…

Pollack's lifting conjecture. There exists a linear combination $T$ of brackets of $\varepsilon_{2i}$, containing at least three $\varepsilon_{2i}$ with $i>1$, such that

Let MZVs be the convergent multiple zeta values, with relations generated by products of their iterated-integral and nested-sum descriptions. The resulting identities are called sh…

Motivic Broadhurst–Kreimer conjecture. The depth-and-weight generating series satisfies