27 problems

Cyclic-sum conjecture. For every integer $n>r$ and every primitive root of unity $\zeta_n$, the following hold. With $r=\sum_{j=0}^{t}d_j+2t$,

Reserved density conjecture.

Let ${\boldsymbol{s}}\in\mathbb N^l$. Let $RJ({\boldsymbol{s}})$ be the reserved divisibility set, and let $RJ_1({\boldsymbol{s}})$ and $RJ_2({\boldsymbol{s}})$ denote the correspo…

Let $l$ be a positive integer and let ${\boldsymbol{s}}\in\mathbb N^l$. Define $J({\boldsymbol{s}}|2)$ to be the set of nonnegative integers $n$ for which the numerator of the mult…

Let $l$ be a positive integer and let ${\boldsymbol{s}}=(s_1,\dots,s_l)$ be a sequence of positive integers. For $n\geq 0$, write the multiple harmonic sum as…

Let ${\mathbf{s}}=(s_1,\ldots,s_l)\in\mathbb N^l$, let $|{\mathbf{s}}|=s_1+\cdots+s_l$, and let $p\ge |{\mathbf{s}}|$ be prime. Multiple-harmonic-sum divisibility conjecture. If…

Let $r\in\mathbb{N}_0$ and $n\in\mathbb{N}$. For any reals $x,y$, let ${\bf p}_{r+1}:=(p_1,p_2,\ldots,p_{r+1})\in\mathbb{N}^{r+1}$ and…

Pilehrood–Pilehrood–Tauraso's conjecture. For any positive integer $n$ with $n>k$, one has

Bernoulli congruences for $R_8^{(m)}(p)$, $R_9^{(m)}(p)$, and $R_{10}^{(m)}(p)$.

Let $k$ be a positive integer and let $p_1$ be a prime greater than $k+1$. For each index $\boldsymbol{k}$, let $H_{p-1}(\boldsymbol{k})$ be the multiple harmonic sum, and let…

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…

Consider the domains and summands described in the source: domains are formed from finite disjoint unions and finite products of sets defined by chains of equalities and inequaliti…

Consider convergent sums of multiple harmonic values that are both $p$-adically and weight-adically convergent. Generation conjecture for multiple harmonic-value relations. Every r…

Let $I\subset\mathcal{P}\times\mathbb{N}$, and let $\widehat{\operatorname{per}}_{\operatorname{har}_I}$ be the period map on weight-adically convergent sums of motivic multiple ha…

Weighted congruence deduction conjecture. Every weighted congruence can be deduced from Theorem.

Mixed congruence reduction conjecture. The mixed congruence holds if and only if, for every $k\in\Z$, the weighted congruence

Fix $N>1$ and work on $\mathbb{P}^{1}-\{0,\mu_N,\infty\}$. Let $p$ range over primes not dividing $N$, let $\alpha\in\mathbb{N}$, and let $\operatorname{har}_{p^{\alpha}}$ denote t…

Let $\mathcal{P}_{\geq 3}$ denote the set of primes at least $3$, let $\alpha\in\mathbb{N}^{\ast}$, and let $I$ be one of $\mathcal{P}_{\geq 3}\times\{\alpha\}$,…

Zhao's conjecture. These two generators should be linearly independent modulo $p$ over ${\mathbb Q}(10)$.

Hoffman's conjecture. Every multiple harmonic sum $S_{p-1}({\bf s})$, or $H_{p-1}({\bf s})$, can be written modulo $p$ as a sum of products of height one sums with rational coeffic…

Reserved-set structure conjecture. If one of the following holds: (i) ${\vec{s}}=1$, (ii) ${\vec{s}}=(1,2,1)$, (iii) ${\vec{s}}=(2r-1,1)$ for some $r\geq1$, or (iv)…