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Fungrim entry: 291569

1L ⁣(s,χ)=n=1μ ⁣(n)χ ⁣(n)ns\frac{1}{L\!\left(s, \chi\right)} = \sum_{n=1}^{\infty} \frac{\mu\!\left(n\right) \chi\!\left(n\right)}{{n}^{s}}
Assumptions:qZ1andχGqandsCandRe ⁣(s)>1q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 1
TeX:
\frac{1}{L\!\left(s, \chi\right)} = \sum_{n=1}^{\infty} \frac{\mu\!\left(n\right) \chi\!\left(n\right)}{{n}^{s}}

q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 1
Definitions:
Fungrim symbol Notation Short description
DirichletLL ⁣(s,χ)L\!\left(s, \chi\right) Dirichlet L-function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
MoebiusMuμ ⁣(n)\mu\!\left(n\right) Möbius function
Powab{a}^{b} Power
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
DirichletGroupGqG_{q} Dirichlet characters with given modulus
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("291569"),
    Formula(Equal(Div(1, DirichletL(s, chi)), Sum(Div(Mul(MoebiusMu(n), chi(n)), Pow(n, s)), Tuple(n, 1, Infinity)))),
    Variables(q, chi, s),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1))))

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2019-08-17 11:32:46.829430 UTC