37 problems

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…

Bailey–Borwein–Girgensohn conjecture. When $p+q>7$ and $p+q$ is even, the double linear sums $S_{p,q}$ are not reducible to zeta values.

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 $r,m\in\mathbb N$, let $f(r,s_{1},s_{2})$ be the index bound occurring in the coefficient recursion, and let $\operatorname{sgn}$ be the signum function defined by … For…

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 $a_i,c_i\in\mathbb{N}^{\ast}$ 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 $\ES_n$ be the $\Q$-vector space spanned by all Euler sums of weight $n$. Broadhurst's dimension conjecture. … This is presented as an Euler-sum analogue of Zagier's dimension…

Recurrence-product conjecture. The sequence satisfies

Minimal-basis conjecture. This set forms a minimal $\mathbf{Q}$-basis for double sums. Such a basis would provide a compact description of the alternating Euler sums appearing in d…

For positive integers $s$ and $t$, define the double Euler sum … and the single sums $\zeta(r)=\sum_{n>0}n^{-r}$. Irreducibility conjecture. The majority of the sums $\zeta(s,t)$ a…

A non-alternating Euler sum has level $l$ and depth $k$, where the level records the parity/sign structure of the sum and the depth is the number of nested summation indices. Reduc…

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…

Euler's parity conjecture. All $S_{p;q}$ with $p+q$ odd can be expressed as a $\mathbb{Q}$-coefficient combination of zeta values.

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…

Higher-order atom conjecture. A result similar to Theorem 1 applies to mixed Euler sums of every higher order; the remaining task is to identify the appropriate atoms at each order…

Uniqueness conjecture. The representation of every mixed Euler sum of order at most $12$ as a rational linear combination of the constants in that list is unique.

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 $a(r,m,\ell)$ 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…