Fungrim entry: 2a48bd

\chi_{{q}_{1} {q}_{2} \, . \, \ell} = \chi_{{q}_{1} \, . \, \ell \bmod {q}_{1}} \chi_{{q}_{2} \, . \, \ell \bmod {q}_{2}}

Assumptions:

{q}_{1} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {q}_{2} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, \max\!\left({q}_{1} {q}_{2}, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {q}_{1}\right) = \gcd\!\left(\ell, {q}_{2}\right) = \gcd\!\left({q}_{1}, {q}_{2}\right) = 1

TeX:

\chi_{{q}_{1} {q}_{2} \, . \, \ell} = \chi_{{q}_{1} \, . \, \ell \bmod {q}_{1}} \chi_{{q}_{2} \, . \, \ell \bmod {q}_{2}}

{q}_{1} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; {q}_{2} \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \ell \in \{1, 2, \ldots, \max\!\left({q}_{1} {q}_{2}, 2\right) - 1\} \;\mathbin{\operatorname{and}}\; \gcd\!\left(\ell, {q}_{1}\right) = \gcd\!\left(\ell, {q}_{2}\right) = \gcd\!\left({q}_{1}, {q}_{2}\right) = 1

Definitions:

Fungrim symbol	Notation	Short description
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Range`	$\{a, a + 1, \ldots, b\}$	Integers between given endpoints
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor

Source code for this entry:

Entry(ID("2a48bd"),
    Formula(Equal(DirichletCharacter(Mul(Subscript(q, 1), Subscript(q, 2)), ell), Mul(DirichletCharacter(Subscript(q, 1), Mod(ell, Subscript(q, 1))), DirichletCharacter(Subscript(q, 2), Mod(ell, Subscript(q, 2)))))),
    Variables(Subscript(q, 1), Subscript(q, 2), ell),
    Assumptions(And(Element(Subscript(q, 1), ZZGreaterEqual(1)), Element(Subscript(q, 2), ZZGreaterEqual(1)), Element(ell, Range(1, Sub(Max(Mul(Subscript(q, 1), Subscript(q, 2)), 2), 1))), Equal(GCD(ell, Subscript(q, 1)), GCD(ell, Subscript(q, 2)), GCD(Subscript(q, 1), Subscript(q, 2)), 1))))

Topics using this entry

Dirichlet characters