`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 | $\chi_{q}(\ell, \cdot)$ | Dirichlet character |

Source code for this entry:

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