20 problems

Recurrence-product conjecture. The sequence satisfies

Let $d$ be a positive integer with $d\ge 3$. Write $1_{d-3}$ for a string of $d-3$ entries equal to $1$, let $\zeta^\sharp$ denote the Euler sharp sum, and let $\zeta(m,n)$ denote…

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…

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 $\mathsf{FMZV}_w$, $\mathsf{FMZV}^{(2)}_w$, $\mathsf{FMSV}_w$, $\mathsf{FES}_w$, $\mathsf{FMTV}_w$, $\mathsf{FMMVe}_w$, $\mathsf{FMMV}_w$,…

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…

Let $a\in\mathbb{N}_0$, $\ell\in\mathbb{N}$, and let $u_j,v_j\in X:=\{1,\bar2,3,\bar4,\dots\}$ for $j=1,\dots,\ell$, with … For $\mathfrak S_\ell$ the symmetric group on $\ell$ let…

Parametric Euler-sum conjecture. These sums can be evaluated in terms of Hurwitz zeta values and digamma functions. This extends the preceding explicitly known cases, but the sourc…

Let $r,m\in\mathbb N$, let $f(r,s_{1},s_{2})$ be the index bound occurring in the coefficient recursion, and let $\delta_{mn}$ denote the Kronecker delta, with $\delta_{mm}=1$ and…

Let $r,m,j\in\mathbb N$, and let $f(r,s_{1},s_{2})$ be the index bound occurring in the coefficient recursion. For $0\leq m\leq f(r,s_{1},s_{2})$ and…

Let $a(r,m,\ell)$ be the coefficients appearing in the expansion under consideration, and let $\operatorname{sgn}(x)$ be the signum function: … For $r,m\in\mathbb N$ with…

Let $a(r,m,\ell)$ be the coefficients appearing in the expansion under consideration. Boundary-sum conjecture. For $r\in\mathbb N$, … and … This is one of four conjectures suggeste…

Let $m$ be a positive integer, and let $\zeta(\overline{r},s)$ denote an alternating double zeta value, while a non-alternating double zeta value has no barred index. Alternating d…

Weight-ten basis conjecture. All non-alternating Euler sums of weight ten can be expressed as a rational linear combination of these seven quantities.

Let $a_i,c_i\in\mathbb{N}^{\ast}$ with $c_i\neq 2$, and define … where $\delta_c=\delta_{c=1}$ equals $1$ if $c=1$ and $0$ otherwise. Define … … and … Here \text{\boldmath 1…

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…

Integral-basis conjecture. There exist $\mathbb{Q}$-linearly independent Euler sums of weight $n$ such that every Euler sum of weight $n$ is a $\mathbb{Z}$-linear combination of th…

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