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 $\mathbb{Z}$ to $\mathbb{C}$.
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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-10-05 13:11:19.856591 UTC