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Fungrim entry: 23256b

lims1(s1)L ⁣(1,χq(1,))=φ ⁣(q)q\lim_{s \to 1} \left(s - 1\right) L\!\left(1, \chi_{q}(1, \cdot)\right) = \frac{\varphi\!\left(q\right)}{q}
Assumptions:qZ1q \in \mathbb{Z}_{\ge 1}
TeX:
\lim_{s \to 1} \left(s - 1\right) L\!\left(1, \chi_{q}(1, \cdot)\right) = \frac{\varphi\!\left(q\right)}{q}

q \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
ComplexLimitlimzaf ⁣(z)\lim_{z \to a} f\!\left(z\right) Limiting value, complex variable
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
DirichletCharacterχq(,)\chi_{q}(\ell, \cdot) Dirichlet character
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
ZZGreaterEqualZn\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))), s, 1), Div(Totient(q), q))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))))

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2019-08-19 14:38:23.809000 UTC