Fungrim entry: a9a405

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

Assumptions:

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1

TeX:

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

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \left|x\right| < 1

Definitions:

Fungrim symbol	Notation	Short description
`Sum`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`Log`	$\log(z)$	Natural logarithm
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("a9a405"),
    Formula(Equal(Sum(Mul(Div(Totient(n), n), Log(Sub(1, Pow(x, n)))), For(n, 1, Infinity)), Div(x, Sub(x, 1)))),
    Variables(x),
    Assumptions(And(Element(x, CC), Less(Abs(x), 1))))

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