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

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

2021-03-15 19:12:00.328586 UTC