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Fungrim entry: ed65c8

Gqprimitive={χ:χGqand[for all d with d{1,2,q1}anddq,there exists a with a{0,1,q1}anda1(modd)andgcd ⁣(a,q)=1andχ ⁣(a)1]}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:qZ1q \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
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
SetBuilder{f ⁣(x):P ⁣(x)}\left\{ f\!\left(x\right) : P\!\left(x\right) \right\} Set comprehension
DirichletGroupGqG_{q} Dirichlet characters with given modulus
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
ZZGreaterEqualZn\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."))

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2019-08-19 14:38:23.809000 UTC