Fungrim home page

Fungrim entry: a9a405

n=1φ(n)nlog ⁣(1xn)=xx1\sum_{n=1}^{\infty} \frac{\varphi(n)}{n} \log\!\left(1 - {x}^{n}\right) = \frac{x}{x - 1}
Assumptions:xCandx<1x \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
Sumnf(n)\sum_{n} f(n) Sum
Totientφ(n)\varphi(n) Euler totient function
Loglog(z)\log(z) Natural logarithm
Powab{a}^{b} Power
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
Absz\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.

2020-01-31 18:09:28.494564 UTC