Fungrim entry: ed65c8

G^{\text{Primitive}}_{q} = \left\{ \chi : \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \left[a \equiv 1 \pmod {d} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, q\right) = 1 \;\mathbin{\operatorname{and}}\; \chi(a) \ne 1 \;\text{ for some } a \in \{0, 1, \ldots, q - 1\} \;\text{ for all } d \in \{1, 2, \ldots, q - 1\} \text{ with } d \mid q\right] \right\}

Assumptions:

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

References:

T. Apostol (1976), Introduction to Analytic Number Theory, Springer. Chapter 8.7.

TeX:

G^{\text{Primitive}}_{q} = \left\{ \chi : \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \left[a \equiv 1 \pmod {d} \;\mathbin{\operatorname{and}}\; \gcd\!\left(a, q\right) = 1 \;\mathbin{\operatorname{and}}\; \chi(a) \ne 1 \;\text{ for some } a \in \{0, 1, \ldots, q - 1\} \;\text{ for all } d \in \{1, 2, \ldots, q - 1\} \text{ with } d \mid q\right] \right\}

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

Definitions:

Fungrim symbol	Notation	Short description
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("ed65c8"),
    Formula(Equal(PrimitiveDirichletCharacters(q), Set(chi, For(chi), And(Element(chi, DirichletGroup(q)), Brackets(All(Exists(And(CongruentMod(a, 1, d), Equal(GCD(a, q), 1), NotEqual(chi(a), 1)), ForElement(a, Range(0, Sub(q, 1)))), ForElement(d, Range(1, Sub(q, 1))), Divides(d, q))))))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))),
    References("T. Apostol (1976), Introduction to Analytic Number Theory, Springer. Chapter 8.7."))

Topics using this entry

Dirichlet characters