Multiple zeta values

Symbol: MultiZetaValue —

\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)

— Multiple zeta value (MZV)

References:

https://www.usna.edu/Users/math/meh/biblio.html

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)

Source code for this entry:

Entry(ID("aab550"),
    SymbolDefinition(MultiZetaValue, MultiZetaValue(Subscript(s, 1), Ellipsis, Subscript(s, k)), "Multiple zeta value (MZV)"),
    References("https://www.usna.edu/Users/math/meh/biblio.html"))

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))))

345c26

\zeta\!\left(2, 1\right) = \sum_{n=1}^{\infty} \frac{H_{n}}{{\left(n + 1\right)}^{2}} = \zeta\!\left(3\right)

TeX:

\zeta\!\left(2, 1\right) = \sum_{n=1}^{\infty} \frac{H_{n}}{{\left(n + 1\right)}^{2}} = \zeta\!\left(3\right)

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
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("345c26"),
    Formula(Equal(MultiZetaValue(2, 1), Sum(Div(HarmonicNumber(n), Pow(Add(n, 1), 2)), For(n, 1, Infinity)), RiemannZeta(3))))

62de01

\zeta\!\left(2, 2\right) = \frac{3}{4} \zeta\!\left(4\right)

TeX:

\zeta\!\left(2, 2\right) = \frac{3}{4} \zeta\!\left(4\right)

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("62de01"),
    Formula(Equal(MultiZetaValue(2, 2), Mul(Div(3, 4), RiemannZeta(4)))))

a5e52e

\zeta\!\left(3, 2\right) = 3 \zeta\!\left(2\right) \zeta\!\left(3\right) - \frac{11}{2} \zeta\!\left(5\right)

TeX:

\zeta\!\left(3, 2\right) = 3 \zeta\!\left(2\right) \zeta\!\left(3\right) - \frac{11}{2} \zeta\!\left(5\right)

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("a5e52e"),
    Formula(Equal(MultiZetaValue(3, 2), Sub(Mul(Mul(3, RiemannZeta(2)), RiemannZeta(3)), Mul(Div(11, 2), RiemannZeta(5))))))

ef2c71

\zeta\!\left(4, 2\right) = {\left(\zeta\!\left(3\right)\right)}^{2} - \frac{4}{3} \zeta\!\left(6\right)

TeX:

\zeta\!\left(4, 2\right) = {\left(\zeta\!\left(3\right)\right)}^{2} - \frac{4}{3} \zeta\!\left(6\right)

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`Pow`	${a}^{b}$	Power
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("ef2c71"),
    Formula(Equal(MultiZetaValue(4, 2), Sub(Pow(RiemannZeta(3), 2), Mul(Div(4, 3), RiemannZeta(6))))))

856317

\zeta\!\left(2, 3\right) = \frac{9}{2} \zeta\!\left(5\right) - 2 \zeta\!\left(2\right) \zeta\!\left(3\right)

TeX:

\zeta\!\left(2, 3\right) = \frac{9}{2} \zeta\!\left(5\right) - 2 \zeta\!\left(2\right) \zeta\!\left(3\right)

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("856317"),
    Formula(Equal(MultiZetaValue(2, 3), Sub(Mul(Div(9, 2), RiemannZeta(5)), Mul(Mul(2, RiemannZeta(2)), RiemannZeta(3))))))

3a5167

\zeta\!\left(3, 3\right) = \frac{1}{2} \left({\left(\zeta\!\left(3\right)\right)}^{2} - \zeta\!\left(6\right)\right)

TeX:

\zeta\!\left(3, 3\right) = \frac{1}{2} \left({\left(\zeta\!\left(3\right)\right)}^{2} - \zeta\!\left(6\right)\right)

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`Pow`	${a}^{b}$	Power
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function

Source code for this entry:

Entry(ID("3a5167"),
    Formula(Equal(MultiZetaValue(3, 3), Mul(Div(1, 2), Sub(Pow(RiemannZeta(3), 2), RiemannZeta(6))))))

ef8b17

\zeta\!\left(s, s\right) = \frac{1}{2} \left({\left(\zeta\!\left(s\right)\right)}^{2} - \zeta\!\left(2 s\right)\right)

Assumptions:

s \in \mathbb{Z}_{\ge 2}

TeX:

\zeta\!\left(s, s\right) = \frac{1}{2} \left({\left(\zeta\!\left(s\right)\right)}^{2} - \zeta\!\left(2 s\right)\right)

s \in \mathbb{Z}_{\ge 2}

Definitions:

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

Source code for this entry:

Entry(ID("ef8b17"),
    Formula(Equal(MultiZetaValue(s, s), Mul(Div(1, 2), Sub(Pow(RiemannZeta(s), 2), RiemannZeta(Mul(2, s)))))),
    Variables(s),
    Assumptions(Element(s, ZZGreaterEqual(2))))

4a23c7

\zeta\!\left(\underbrace{2, \ldots, 2}_{n \text{ times}}\right) = \frac{{\pi}^{2 n}}{\left(2 n + 1\right)!}

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\zeta\!\left(\underbrace{2, \ldots, 2}_{n \text{ times}}\right) = \frac{{\pi}^{2 n}}{\left(2 n + 1\right)!}

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`Pow`	${a}^{b}$	Power
`Pi`	$\pi$	The constant pi (3.14...)
`Factorial`	$n !$	Factorial
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("4a23c7"),
    Formula(Equal(MultiZetaValue(Repeat(2, n)), Div(Pow(Pi, Mul(2, n)), Factorial(Add(Mul(2, n), 1))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

da71d3

\zeta\!\left(a, b\right) + \zeta\!\left(b, a\right) = \zeta\!\left(a\right) \zeta\!\left(b\right) - \zeta\!\left(a + b\right)

Assumptions:

a \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z}_{\ge 2}

TeX:

\zeta\!\left(a, b\right) + \zeta\!\left(b, a\right) = \zeta\!\left(a\right) \zeta\!\left(b\right) - \zeta\!\left(a + b\right)

a \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; b \in \mathbb{Z}_{\ge 2}

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("da71d3"),
    Formula(Equal(Add(MultiZetaValue(a, b), MultiZetaValue(b, a)), Sub(Mul(RiemannZeta(a), RiemannZeta(b)), RiemannZeta(Add(a, b))))),
    Variables(a, b),
    Assumptions(And(Element(a, ZZGreaterEqual(2)), Element(b, ZZGreaterEqual(2)))))

70c42b

\zeta\!\left(\underbrace{3, 1, \ldots, 3, 1}_{\left(3, 1\right) \; n \text{ times}}\right) = \frac{1}{2 n + 1} \zeta\!\left(\underbrace{2, \ldots, 2}_{2 n \text{ times}}\right)

Assumptions:

n \in \mathbb{Z}_{\ge 1}

TeX:

\zeta\!\left(\underbrace{3, 1, \ldots, 3, 1}_{\left(3, 1\right) \; n \text{ times}}\right) = \frac{1}{2 n + 1} \zeta\!\left(\underbrace{2, \ldots, 2}_{2 n \text{ times}}\right)

n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`MultiZetaValue`	$\zeta\!\left({s}_{1}, \ldots, {s}_{k}\right)$	Multiple zeta value (MZV)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("70c42b"),
    Formula(Equal(MultiZetaValue(Repeat(3, 1, n)), Mul(Div(1, Add(Mul(2, n), 1)), MultiZetaValue(Repeat(2, Mul(2, n)))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))))

Definitions

Sum representations

Specific values

Families of closed forms

Relations