Fungrim entry: 94a39f

\zeta\!\left({s}_{1}, {s}_{2}, \ldots, {s}_{k}\right) = \sum_{\textstyle{n \in {\mathbb{Z}}^{k} \atop {n}_{1} > {n}_{2} > \ldots > {n}_{k} > 0}} \prod_{i=1}^{k} \frac{1}{{n}_{i}^{{s}_{i}}}

Assumptions:

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; s \in {\left(\mathbb{Z}_{\ge 1}\right)}^{k} \;\mathbin{\operatorname{and}}\; \sum_{i=1}^{k} {s}_{i} > k

TeX:

\zeta\!\left({s}_{1}, {s}_{2}, \ldots, {s}_{k}\right) = \sum_{\textstyle{n \in {\mathbb{Z}}^{k} \atop {n}_{1} > {n}_{2} > \ldots > {n}_{k} > 0}} \prod_{i=1}^{k} \frac{1}{{n}_{i}^{{s}_{i}}}

k \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; s \in {\left(\mathbb{Z}_{\ge 1}\right)}^{k} \;\mathbin{\operatorname{and}}\; \sum_{i=1}^{k} {s}_{i} > k

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`Sum`	$\sum_{n} f(n)$	Sum
`Product`	$\prod_{n} f(n)$	Product
`Pow`	${a}^{b}$	Power
`ZZ`	$\mathbb{Z}$	Integers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("94a39f"),
    Formula(Equal(MultiZetaValue(Step(Subscript(s, i), For(i, 1, k))), Sum(Product(Div(1, Pow(Subscript(n, i), Subscript(s, i))), For(i, 1, k)), ForElement(n, Pow(ZZ, k)), Greater(Step(Subscript(n, i), For(i, 1, k)), 0)))),
    Variables(k, s),
    Assumptions(And(Element(k, ZZGreaterEqual(1)), Element(s, Pow(ZZGreaterEqual(1), k)), Greater(Sum(Subscript(s, i), For(i, 1, k)), k))))

Topics using this entry

Multiple zeta values