Fungrim entry: a7d592

\chi = {\chi}_{0} {\chi}_{1} \;\text{ for some } \left(d, {\chi}_{0}\right) \text{ with } d \in \{1, 2, \ldots, q\} \;\mathbin{\operatorname{and}}\; d \mid q \;\mathbin{\operatorname{and}}\; {\chi}_{0} \in G^{\text{Primitive}}_{d}\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1}

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

TeX:

\chi = {\chi}_{0} {\chi}_{1} \;\text{ for some } \left(d, {\chi}_{0}\right) \text{ with } d \in \{1, 2, \ldots, q\} \;\mathbin{\operatorname{and}}\; d \mid q \;\mathbin{\operatorname{and}}\; {\chi}_{0} \in G^{\text{Primitive}}_{d}\; \text{ where } {\chi}_{1} = \chi_{q \, . \, 1}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q}

Definitions:

Fungrim symbol	Notation	Short description
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`PrimitiveDirichletCharacters`	$G^{\text{Primitive}}_{q}$	Primitive Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus

Source code for this entry:

Entry(ID("a7d592"),
    Formula(Where(Exists(Equal(chi, Mul(Subscript(chi, 0), Subscript(chi, 1))), For(Tuple(d, Subscript(chi, 0))), And(Element(d, Range(1, q)), Divides(d, q), Element(Subscript(chi, 0), PrimitiveDirichletCharacters(d)))), Equal(Subscript(chi, 1), DirichletCharacter(q, 1)))),
    Variables(q, Subscript(chi, 0)),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

Topics using this entry

Dirichlet characters