Fungrim entry: 1a907e

\sum_{n=1}^{\infty} \frac{\varphi(n)}{{n}^{s}} = \frac{\zeta\!\left(s - 1\right)}{\zeta\!\left(s\right)}

Assumptions:

s \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\!\left(s\right)}

s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 2

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RiemannZeta`	$\zeta\!\left(s\right)$	Riemann zeta function
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	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

Totient function

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

2021-03-15 19:12:00.328586 UTC