Fungrim entry: e0a6a2

Symbol: RiemannZeta —

\zeta\!\left(s\right)

— Riemann zeta function

The Riemann zeta function

\zeta\!\left(s\right)

is a function of one complex variable

s

. It is a meromorphic function with a pole at

s = 1

. The following table lists all conditions such that RiemannZeta(s) is defined in Fungrim.

Domain	Codomain
Numbers
$s \in \left(1, \infty\right)$	$\zeta\!\left(s\right) \in \left(1, \infty\right)$
$s \in \mathbb{R} \setminus \left\{1\right\}$	$\zeta\!\left(s\right) \in \mathbb{R}$
$s \in \mathbb{C} \setminus \left\{1\right\}$	$\zeta\!\left(s\right) \in \mathbb{C}$
Infinities
$s \in \left\{1\right\}$	$\zeta\!\left(s\right) \in \left\{{\tilde \infty}\right\}$
$s \in \left\{\infty\right\}$	$\zeta\!\left(s\right) \in \left\{1\right\}$
Formal power series
$s \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{R}[[x]]$
$s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; [{x}^{0}] s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{C}[[x]]$
$s \in \mathbb{R}[[x]] \;\mathbin{\operatorname{and}}\; s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{R}(\!(x)\!)$
$s \in \mathbb{C}[[x]] \;\mathbin{\operatorname{and}}\; s \ne 1$	$\zeta\!\left(s\right) \in \mathbb{C}(\!(x)\!)$

Table data:

\left(P, Q\right)

such that

\left(P\right) \;\implies\; \left(Q\right)

Definitions:

Fungrim symbol	Notation	Short description
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers
`CC`	$\mathbb{C}$	Complex numbers
`UnsignedInfinity`	${\tilde \infty}$	Unsigned infinity
`PowerSeries`	$K[[x]]$	Formal power series
`LaurentSeries`	$K(\!(x)\!)$	Formal Laurent series

Source code for this entry:

Entry(ID("e0a6a2"),
    SymbolDefinition(RiemannZeta, RiemannZeta(s), "Riemann zeta function"),
    Description("The Riemann zeta function", RiemannZeta(s), "is a function of one complex variable", s, ". It is a meromorphic function with a pole at", Equal(s, 1), ".", "The following table lists all conditions such that", SourceForm(RiemannZeta(s)), "is defined in Fungrim."),
    Table(TableRelation(Tuple(P, Q), Implies(P, Q)), TableHeadings(Description("Domain"), Description("Codomain")), List(TableSection("Numbers"), Tuple(Element(s, OpenInterval(1, Infinity)), Element(RiemannZeta(s), OpenInterval(1, Infinity))), Tuple(Element(s, SetMinus(RR, Set(1))), Element(RiemannZeta(s), RR)), Tuple(Element(s, SetMinus(CC, Set(1))), Element(RiemannZeta(s), CC)), TableSection("Infinities"), Tuple(Element(s, Set(1)), Element(RiemannZeta(s), Set(UnsignedInfinity))), Tuple(Element(s, Set(Infinity)), Element(RiemannZeta(s), Set(1))), TableSection("Formal power series"), Tuple(And(Element(s, PowerSeries(RR, x)), NotEqual(SeriesCoefficient(s, x, 0), 1)), Element(RiemannZeta(s), PowerSeries(RR, x))), Tuple(And(Element(s, PowerSeries(CC, x)), NotEqual(SeriesCoefficient(s, x, 0), 1)), Element(RiemannZeta(s), PowerSeries(CC, x))), Tuple(And(Element(s, PowerSeries(RR, x)), NotEqual(s, 1)), Element(RiemannZeta(s), LaurentSeries(RR, x))), Tuple(And(Element(s, PowerSeries(CC, x)), NotEqual(s, 1)), Element(RiemannZeta(s), LaurentSeries(CC, x))))))

Fungrim entry: e0a6a2

Topics using this entry