# Fungrim entry: 2a48bd

$\chi_{{q}_{1} {q}_{2}}(\ell, \cdot) = \chi_{{q}_{1}}(\ell \bmod {q}_{1}, \cdot) \chi_{{q}_{2}}(\ell \bmod {q}_{2}, \cdot)$
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, \cdot) = \chi_{{q}_{1}}(\ell \bmod {q}_{1}, \cdot) \chi_{{q}_{2}}(\ell \bmod {q}_{2}, \cdot)

{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$\chi_{q}(\ell, \cdot)$ Dirichlet character
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
ZZBetween$\{a, a + 1, \ldots b\}$ Integers between a and b inclusive
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, ZZBetween(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

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

2019-08-17 11:32:46.829430 UTC