480 problems

Let $\mathcal{Z}_k$ denote the $\mathbb{Q}$-space of multiple zeta values of weight $k$. Zagier's dimension conjecture. The generating series of their dimensions should satisfy … T…

Kaneko–Zagier's finite double-zeta conjecture. The two definitions coincide:

Let $k$ be a positive integer. A multiple zeta value of weight $k$ is a value … where $s_1,\ldots,s_l$ are positive integers, $s_l>1$, and $s_1+\cdots+s_l=k$. Hoffman's conjecture.…

Multiple zeta values are real numbers obtained by shuffle-regularised iterated sums or integrals, and motivic multiple zeta values are their algebraic counterparts in the ring…

Let $N(w,d)$ denote the number of primitive multiple zeta values (MZVs) of weight $w$ and depth $d$, and let $x$ and $y$ be formal variables. Broadhurst–Kreimer conjecture. The gen…

Let $G$ be a subdivergence-free graph with $n$ loops, let $I^{\mathrm{res}}_G$ denote its Feynman residue, and let $Z_n$ be the zig-zag graph with $n$ loops. Hepp's zig-zag graph c…

Zero-measure conjecture.

Kaneko–Yamamoto conjecture. Any linear dependency of multiple zeta values over $\mathbb Q$ is deduced from this equation; equivalently, the integral-series identity is sufficient t…

For every positive integer $k$, consider the normalized odd zeta value $\zeta(2k+1)/\pi^{2k+1}$. Transcendence conjecture. All numbers … are transcendental. The source presents thi…

Let $S_{p,q}:=\displaystyle\!sum_{n=1}^{\infty}H_n^{(p)}/n^q$ for $p,q\in\mathbb{N}$ and $q>1$, where $H_n^{(p)}:=\displaystyle\!sum_{k=1}^n1/k^p$ is the generalized harmonic numbe…

Let $\Phi^{\mathfrak{m}}(x,y)$ be the motivic version of the modified Drinfeld associator, and let $\Phi(x,y)$ be the corresponding modified Drinfeld associator defined in the sour…

Kaneko's conjecture. The map

Let ${\bf a}$ be an admissible composition whose entries belong to $\{2,3\}$, and let $s$ be the number of entries of ${\bf a}$ equal to $3$. Write $\zeta({\bf a})$ for the multipl…

Let $\bk$ be an index, let $\zeta_{\widehat{\mathcal{A}}}(\bk)$ denote the corresponding $\bsp$-adic finite multiple zeta value, and let the $t$-adic symmetric multiple zeta value…

Let $\operatorname{\mathbf{IKM}}(a,b;n)$ denote the corresponding on-shell Bessel moment, and let $\zeta_{7,1}(2)$ denote the multiple zeta value appearing in the Broadhurst–Mellit…

Thakur's basis conjecture. The values $\zeta_{A}(\mathfrak{s})$ for $\mathfrak{s} \in \mathcal{I}^{\mathrm{T}}$ form a basis of $\mathcal{Z}_{k}$.

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

Let $C_n(\lambda)$ be the polynomial coefficient arising in the asymptotic expansion studied in the paper, and let ${\mathfrak{Z}}_n^{\mathrm{sv}}$ denote the space of single-value…

Let $\mathcal{Z}_k$ be the weight-$k$ space of ordinary multiple zeta values. For a sequence $(k_1,\ldots,k_r)$, call it Lyndon when every proper right subsequence is greater than…

Let $\mathcal T$ be the $\mathbb Q$-subalgebra of $\mathbb R$ generated by $1$ and the multiple $t$-values, and let $\mathcal T_n$ be the subspace of $\mathcal T$ spanned by all mu…

Let $\mathfrak{g}^{\mathfrak{m}}$ be the motivic Lie algebra of multiple zeta values, equipped with its depth filtration $D$, and let…

Let ${\rm Cor}_{\cal M}$ be a motivic correlator defined over a number field $F$, and suppose that in the situation described in the source the correspondence principle holds and t…

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

Thakur's conjecture. Multiple zeta values in positive characteristic are linearly independent over $\mathbb F_q$. The conjecture was formulated as a fundamental problem in function…

Let $\mathcal{Z}_q$ be the vector space of Schlesinger–Zudilin $q$-multiple zeta values, equipped with the stuffle product $\ast$ and duality involution $\tau$. Bachmann's conjectu…