Fungrim entry: e810d8

Symbol: DirichletGroup —

G_{q}

— Dirichlet characters with given modulus

DirichletGroup(q), rendered as

G_{q}

, represents the set of Dirichlet characters modulo

q

, given

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

Dirichlet characters can be defined axiomatically as functions from

\mathbb{Z}

\mathbb{C}

satisfying the properties in formulas 1c3957, 0851cf, and afd0c5.

In this definition, the modulus

q

is not an attribute of the character; for example the character giving the sequence

\left[0, 1, 0, 1, \ldots\right]

is an element of both

G_{2}

and

G_{4}

A more explicit construction of the characters is possible using the Conrey numbering scheme, which is implemented by DirichletCharacter.

Definitions:

Fungrim symbol	Notation	Short description
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("e810d8"),
    SymbolDefinition(DirichletGroup, DirichletGroup(q), "Dirichlet characters with given modulus"),
    Description(SourceForm(DirichletGroup(q)), ", rendered as", DirichletGroup(q), ", represents the set of Dirichlet characters modulo", q, ", given", Element(q, ZZGreaterEqual(1)), "."),
    Description("Dirichlet characters can be defined axiomatically as functions from", ZZ, "to", CC, "satisfying the properties in formulas", EntryReference("1c3957"), ", ", EntryReference("0851cf"), ", and", EntryReference("afd0c5"), "."),
    Description("In this definition, the modulus", q, "is not an attribute of the character; for example", "the character giving the sequence", List(0, 1, 0, 1, Ellipsis), "is an element of both", DirichletGroup(2), "and", DirichletGroup(4), "."),
    Description("A more explicit construction of the characters is possible using the Conrey numbering scheme, which is implemented by", SourceForm(DirichletCharacter), "."))

Topics using this entry

Dirichlet characters