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

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