Fungrim entry: 23256b

\lim_{s \to 1} \left(s - 1\right) L\!\left(1, \chi_{q \, . \, 1}\right) = \frac{\varphi(q)}{q}

Assumptions:

q \in \mathbb{Z}_{\ge 1}

TeX:

\lim_{s \to 1} \left(s - 1\right) L\!\left(1, \chi_{q \, . \, 1}\right) = \frac{\varphi(q)}{q}

q \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("23256b"),
    Formula(Equal(ComplexLimit(Mul(Sub(s, 1), DirichletL(1, DirichletCharacter(q, 1))), For(s, 1)), Div(Totient(q), q))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))))

Topics using this entry

Dirichlet L-functions

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

2021-03-15 19:12:00.328586 UTC