Fungrim entry: 7f5468

\sum_{n=1}^{\infty} \frac{\varphi(n) {q}^{n}}{1 - {q}^{n}} = \frac{q}{{\left(1 - q\right)}^{2}}

Assumptions:

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

TeX:

\sum_{n=1}^{\infty} \frac{\varphi(n) {q}^{n}}{1 - {q}^{n}} = \frac{q}{{\left(1 - q\right)}^{2}}

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

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
`CC`	$\mathbb{C}$	Complex numbers
`Abs`	$\left\|z\right\|$	Absolute value

Source code for this entry:

Entry(ID("7f5468"),
    Formula(Equal(Sum(Div(Mul(Totient(n), Pow(q, n)), Sub(1, Pow(q, n))), For(n, 1, Infinity)), Div(q, Pow(Sub(1, q), 2)))),
    Variables(q),
    Assumptions(And(Element(q, CC), Less(Abs(q), 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