Fungrim entry: d9a187

\chi(n) = \chi_{q \, . \, \ell}\!\left(n\right)\; \text{ where } \chi = \chi_{q \, . \, \ell}

This is simply a syntactical definition of character evaluation.

Assumptions:

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

TeX:

\chi(n) = \chi_{q \, . \, \ell}\!\left(n\right)\; \text{ where } \chi = \chi_{q \, . \, \ell}

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

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
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("d9a187"),
    Formula(Where(Equal(chi(n), DirichletCharacter(q, ell, n)), Equal(chi, DirichletCharacter(q, ell)))),
    Description("This is simply a syntactical definition of character evaluation."),
    Variables(q, ell, n),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(ell, Range(1, Sub(Max(q, 2), 1))), Equal(GCD(ell, q), 1), Element(n, ZZ))))

Topics using this entry

Dirichlet characters