Fungrim entry: c7e2fb

Symbol: DirichletCharacter —

\chi_{q \, . \, \ell}

— Dirichlet character

DirichletCharacter(q, ell), rendered as

\chi_{q \, . \, \ell}

, represents the Dirichlet character with Conrey label

\left(q, \ell\right)

A character represents an object

\chi

that can be called (

\chi(n)

) as a function from

\mathbb{Z}

\mathbb{C}

DirichletCharacter(q, ell, n), rendered as

\chi_{q \, . \, \ell}\!\left(n\right)

, represents the Dirichlet character with Conrey label

\left(q, \ell\right)

evaluated at the integer

n

The Conrey label consists of integers

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

and

\ell \in \{1, 2, \ldots, \max\!\left(q, 2\right) - 1\}

such that

\gcd\!\left(\ell, q\right) = 1

. In this scheme

\chi_{q \, . \, 1}

always represents the trivial/principal character (taking only values 0 and 1) modulo

q

. Non-principal characters are defined by 4cf4e4 when

q

is an odd prime power, by fc4f6a and 03fbe8 when

q

is an even prime power, and in general by factoring

q

into prime powers using 2a48bd.

References:

http://www.lmfdb.org/Character/Labels

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZ`	$\mathbb{Z}$	Integers
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("c7e2fb"),
    SymbolDefinition(DirichletCharacter, DirichletCharacter(q, ell), "Dirichlet character"),
    Description(SourceForm(DirichletCharacter(q, ell)), ", rendered as", DirichletCharacter(q, ell), ", represents the Dirichlet character with Conrey label", Tuple(q, ell), "."),
    Description("A character represents an object", chi, " that can be called (", chi(n), ") as a function from", ZZ, "to", CC, "."),
    Description(SourceForm(DirichletCharacter(q, ell, n)), ", rendered as", DirichletCharacter(q, ell, n), ", represents the Dirichlet character with Conrey label", Tuple(q, ell), "evaluated at the integer", n, "."),
    Description("The Conrey label consists of integers", Element(q, ZZGreaterEqual(1)), "and", Element(ell, Range(1, Sub(Max(q, 2), 1))), "such that", Equal(GCD(ell, q), 1), ". ", "In this scheme", DirichletCharacter(q, 1), "always represents the trivial/principal character (taking only values 0 and 1) modulo", q, ". ", "Non-principal characters are defined by", EntryReference("4cf4e4"), "when", q, "is an odd prime power, by", EntryReference("fc4f6a"), "and", EntryReference("03fbe8"), "when", q, "is an even prime power, and in general by factoring", q, "into prime powers using", EntryReference("2a48bd"), "."),
    References("http://www.lmfdb.org/Character/Labels"))

Topics using this entry

Dirichlet characters