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Fungrim entry: a7d592

there exists (d,χ0) with d{1,2,q}anddqandχ0Gdprimitiveandχ=χ0χ1   where χ1=χq(1,)\text{there exists } \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_{d}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, \chi = {\chi}_{0} {\chi}_{1}\; \text{ where } {\chi}_{1} = \chi_{q}(1, \cdot)
Assumptions:qZ1andχGqq \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
TeX:
\text{there exists } \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_{d}^{\text{primitive}} \,\mathbin{\operatorname{and}}\, \chi = {\chi}_{0} {\chi}_{1}\; \text{ where } {\chi}_{1} = \chi_{q}(1, \cdot)

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q}
Definitions:
Fungrim symbol Notation Short description
ZZBetween{a,a+1,b}\{a, a + 1, \ldots b\} Integers between a and b inclusive
PrimitiveDirichletCharactersGqprimitiveG_{q}^{\text{primitive}} Primitive Dirichlet characters with given modulus
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("a7d592"),
    Formula(Where(Exists(Tuple(d, Subscript(chi, 0)), And(Element(d, ZZBetween(1, q)), Divides(d, q), Element(Subscript(chi, 0), PrimitiveDirichletCharacters(d)), Equal(chi, Mul(Subscript(chi, 0), Subscript(chi, 1))))), Equal(Subscript(chi, 1), DirichletCharacter(q, 1)))),
    Variables(q, Subscript(chi, 0)),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)))))

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2019-08-17 11:32:46.829430 UTC