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Fungrim entry: a9a405

n=1φ ⁣(n)nlog ⁣(1xn)=xx1\sum_{n=1}^{\infty} \frac{\varphi\!\left(n\right)}{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\!\left(n\right)}{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\!\left(n\right) Sum
Totientφ ⁣(n)\varphi\!\left(n\right) Euler totient function
Loglog ⁣(z)\log\!\left(z\right) 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)))), Tuple(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.

2019-09-19 20:12:49.583742 UTC