# 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))))))

## 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