# Fungrim entry: ed65c8

$G_{q}^{\text{primitive}} = \left\{ \chi : \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \left[\text{for all } d \text{ with } d \in \{1, 2, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, d \mid q, \text{there exists } a \text{ with } a \in \{0, 1, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, a \equiv 1 \pmod {d} \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, q\right) = 1 \,\mathbin{\operatorname{and}}\, \chi\!\left(a\right) \ne 1\right] \right\}$
Assumptions:$q \in \mathbb{Z}_{\ge 1}$
References:
• T. Apostol (1976), Introduction to Analytic Number Theory, Springer. Chapter 8.7.
TeX:
G_{q}^{\text{primitive}} = \left\{ \chi : \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \left[\text{for all } d \text{ with } d \in \{1, 2, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, d \mid q, \text{there exists } a \text{ with } a \in \{0, 1, \ldots q - 1\} \,\mathbin{\operatorname{and}}\, a \equiv 1 \pmod {d} \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, q\right) = 1 \,\mathbin{\operatorname{and}}\, \chi\!\left(a\right) \ne 1\right] \right\}

q \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
PrimitiveDirichletCharacters$G_{q}^{\text{primitive}}$ Primitive Dirichlet characters with given modulus
SetBuilder$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$ Set comprehension
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
ZZBetween$\{a, a + 1, \ldots b\}$ Integers between a and b inclusive
GCD$\gcd\!\left(a, b\right)$ Greatest common divisor
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("ed65c8"),
Formula(Equal(PrimitiveDirichletCharacters(q), SetBuilder(chi, chi, And(Element(chi, DirichletGroup(q)), Brackets(ForAll(d, And(Element(d, ZZBetween(1, Sub(q, 1))), Divides(d, q)), Exists(a, And(Element(a, ZZBetween(0, Sub(q, 1))), CongruentMod(a, 1, d), Equal(GCD(a, q), 1), Unequal(chi(a), 1))))))))),
Variables(q),
Assumptions(Element(q, ZZGreaterEqual(1))),
References("T. Apostol (1976), Introduction to Analytic Number Theory, Springer. Chapter 8.7."))

## Topics using this entry

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

2019-08-19 14:38:23.809000 UTC