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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}\right) = \frac{\varphi(q)}{q}
Assumptions:qZ1q \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
ComplexLimitlimzaf(z)\lim_{z \to a} f(z) Limiting value, complex variable
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
DirichletCharacterχq.\chi_{q \, . \, \ell} Dirichlet character
Totientφ(n)\varphi(n) 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))), For(s, 1)), Div(Totient(q), q))),
    Variables(q),
    Assumptions(Element(q, ZZGreaterEqual(1))))

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2019-10-05 13:11:19.856591 UTC