# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC