32 problems

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

Kaneko's conjecture. The map

Let $a$ be an odd positive integer, and let $T_\mathcal{A}(\{1\}^{a})$ denote the finite multiple mixed value associated with the index consisting of $a$ copies of $1$. Let $Z(a)$…

Dimension conjecture. The sequence $(d_w)_{w=0}^{\infty}$ satisfies

Let $\mathsf{FMT}_w$ be the $\mathbb{Q}$-vector space generated by finite multiple $T$-values of weight $w$. Finite multiple $T$-value dimension conjecture. For all $k\geq1$, … Thi…

Let $S_{\mathcal A}$ and $S_\shuffle^{\mathcal S}$ denote the finite and shuffle-regularized symmetric $S$-values, respectively. Even all-ones S-value conjecture. For every even…

Let $T_{\mathcal A}$ and $T_\shuffle^{\mathcal S}$ denote the finite and shuffle-regularized symmetric multiple $T$-values, respectively; let $S_{\mathcal A}$ and…

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

For $w\in\mathbb{N}$, let $\mathsf{FES}_{w}$ and $\mathsf{ES}_w$ be the $\mathbb{Q}$-vector spaces generated by finite Euler sums and Euler sums of weight $w$, respectively. Let…

Let $F_w$ denote the $w$-th Fibonacci number, and let the symbols $\mathsf{FMZV}^{(2)}_w$, $\mathsf{FMMVe}_w$, $\mathsf{FMMV}_w$, $\mathsf{FES}_w$, $\mathsf{FMMVo}_w$,…

For each weight $w$, let $\mathsf{FMZV}_w$, $\mathsf{FMZV}^{(2)}_w$, $\mathsf{FMSV}_w$, $\mathsf{FES}_w$, $\mathsf{FMTV}_w$, $\mathsf{FMMVe}_w$, $\mathsf{FMMV}_w$,…

For each weight $w\in\mathbb{N}$, let $\mathsf{FMZV}^{(2)}_w$ be the $\mathbb{Q}$-vector space generated by level-two finite multiple zeta values of weight $w$, and let…

Let $\bbN$ denote the positive integers. For each weight $w\in\mathbb{N}$, let $\mathsf{FES}_w$ be the $\mathbb{Q}$-vector space generated by finite Euler sums of weight $w$, and l…

For $r\geq 1$ and $0\leq a\leq r$, consider all indices $(k_1,\ldots,k_r)$ with $k_i\in\{1,2\}$ and exactly $a$ entries equal to $2$. Let $\#\{i\mid i\text{ odd},\ k_i=2\}$ count t…

Let $\mathcal{Z}_{\mathcal{A}}^{(1)}$ be the $\mathbb{Q}$-vector space of ordinary finite multiple zeta values, and let $\zeta_{\mathcal{A}}^{(2)}(k_1,\ldots,k_r)$ denote a level-t…

Let $\mathcal{Z}_{\mathcal{A}}^{(2)}$ and $\mathcal{E}$ be the $\mathbb{Q}$-vector spaces of level-two finite multiple zeta values and finite Euler sums, respectively. For an index…

For an index $\boldsymbol{k}=(k_1,\ldots,k_r)\in\mathbb{I}^+$ with $r\neq1$, let $\omega_{\mathcal{A}}(\boldsymbol{k})$ and $\omega_{\mathcal{S}}(\boldsymbol{k})$ be the finite and…

For a positive integer $n$, let $\mathcal{A}_n$-MZVs and $\mathcal{S}_n$-MZVs denote the finite and symmetric multiple zeta values defined in the paper, respectively. Finite–symmet…

For $n_{1},\dots,n_{d}\in\mathbb{Q}$, define … and let $W_{k}^{\star}(n_{1},\dots,n_{d})$ be the analogous sum with $\zeta_{\mathcal{A}}^{\star}$. For $k,r\in\mathbb{Z}_{\geq1}$ an…

Let $W_{k}(n_{1},\dots,n_{d})$ and $W_{k}^{\star}(n_{1},\dots,n_{d})$ denote the weighted sums of finite multiple zeta values and finite star multiple zeta values, respectively, de…

Define the weighted sums of finite multiple zeta values by … and define $W_{k}^{\star}(n_{1},\dots,n_{d})$ similarly with $\zeta_{\mathcal{A}}^{\star}$ in place of…

Let $m,n\in\mathbb{N}$, and let $\gb_a$ denote the $\mathcal{A}$-Bernoulli numbers. Write $a_1+\dots+a_l\vdash n$ for the summation condition used in the source, and let…

Let $\mathbb{N}$ denote the positive integers. For each $m,n\in\mathbb{N}$, let $R_n^{(m,1)}$ and $S_n^{(m,1)}$ be the finite multiple harmonic sum quantities defined in the source…

Let $\operatorname{red}_{\zeta}$ be the map obtained by reducing $p$-adic multiple zeta values modulo sufficiently large primes, from the $\mathbb{Q}$-algebra of $p$-adic multiple…

Let $\mathfrak H^1$ be the relevant noncommutative polynomial algebra of words, let $z=x+y$, and let $\partial_l$ be the derivation indexed by $l\in\mathbb Z_{\geq1}$. For…