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Fungrim entry: 1a907e

n=1φ ⁣(n)ns=ζ ⁣(s1)ζ ⁣(s)\sum_{n=1}^{\infty} \frac{\varphi\!\left(n\right)}{{n}^{s}} = \frac{\zeta\!\left(s - 1\right)}{\zeta\!\left(s\right)}
Assumptions:sCandRe ⁣(s)>2s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 2
TeX:
\sum_{n=1}^{\infty} \frac{\varphi\!\left(n\right)}{{n}^{s}} = \frac{\zeta\!\left(s - 1\right)}{\zeta\!\left(s\right)}

s \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(s\right) > 2
Definitions:
Fungrim symbol Notation Short description
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
Powab{a}^{b} Power
Infinity\infty Positive infinity
RiemannZetaζ ⁣(s)\zeta\!\left(s\right) Riemann zeta function
CCC\mathbb{C} Complex numbers
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("1a907e"),
    Formula(Equal(Sum(Div(Totient(n), Pow(n, s)), Tuple(n, 1, Infinity)), Div(RiemannZeta(Sub(s, 1)), RiemannZeta(s)))),
    Variables(s),
    Assumptions(And(Element(s, CC), Greater(Re(s), 2))))

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

2019-08-17 11:32:46.829430 UTC