# 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

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