Fungrim symbol: RiemannZeta

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))))))

Topics using this symbol

The symbol RiemannZeta appears in 22 topics:

Riemann zeta function (in 29 entries)
Hurwitz zeta function (in 18 entries)
Specific values of the digamma function (in 10 entries)
Multiple zeta values (in 8 entries)
Zeros of the Riemann zeta function (in 5 entries)
Integrals of Jacobi theta functions (in 5 entries)
Eisenstein series (in 5 entries)
Dirichlet L-functions (in 5 entries)
Digamma function (in 5 entries)
Riemann hypothesis (in 3 entries)
Jacobi theta functions (in 3 entries)
Totient function (in 2 entries)
Pi (in 2 entries)
Euler's constant (in 2 entries)
Catalan's constant (in 2 entries)
Barnes G-function (in 2 entries)
Stieltjes constants (in 1 entries)
Sinc function (in 1 entries)
Modular transformations (in 1 entries)
Greatest common divisor (in 1 entries)
Gamma function (in 1 entries)
Bernoulli numbers and polynomials (in 1 entries)

Entries using this symbol

The symbol RiemannZeta appears in 101 entries: