Fungrim home page

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(a) \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(a) \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
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
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), Set(chi, For(chi), And(Element(chi, DirichletGroup(q)), Brackets(ForAll(d, And(Element(d, Range(1, Sub(q, 1))), Divides(d, q)), Exists(a, And(Element(a, Range(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.

2019-10-05 13:11:19.856591 UTC