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Fungrim entry: 1a907e

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

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

Topics using this entry

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

2019-10-05 13:11:19.856591 UTC