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Fungrim entry: ed65c8

Gqprimitive={χ:χGq  and  [a1(modd)  and  gcd ⁣(a,q)=1  and  χ(a)1   for some a{0,1,,q1}   for all d{1,2,,q1} with dq]}G_{q}^{\text{primitive}} = \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: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[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
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
DirichletGroupGqG_{q} Dirichlet characters with given modulus
GCDgcd ⁣(a,b)\gcd\!\left(a, b\right) Greatest common divisor
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
ZZGreaterEqualZn\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."))

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2020-04-08 16:14:44.404316 UTC