Fungrim entry: 3f5711

\varphi(n) = \sum_{k=1}^{n} \gcd\!\left(n, k\right) {e}^{2 \pi i k / n}

Assumptions:

n \in \mathbb{Z}_{\ge 0}

TeX:

\varphi(n) = \sum_{k=1}^{n} \gcd\!\left(n, k\right) {e}^{2 \pi i k / n}

n \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Totient`	$\varphi(n)$	Euler totient function
`Sum`	$\sum_{n} f(n)$	Sum
`GCD`	$\gcd\!\left(a, b\right)$	Greatest common divisor
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`ConstI`	$i$	Imaginary unit
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("3f5711"),
    Formula(Equal(Totient(n), Sum(Mul(GCD(n, k), Exp(Div(Mul(Mul(Mul(2, Pi), ConstI), k), n))), For(k, 1, n)))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(0))))

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