Fungrim entry: 93a877

\varphi(n) = \sum_{k=1}^{n} \gcd\!\left(n, k\right) \cos\!\left(\frac{2 \pi k}{n}\right)

Assumptions:

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

TeX:

\varphi(n) = \sum_{k=1}^{n} \gcd\!\left(n, k\right) \cos\!\left(\frac{2 \pi k}{n}\right)

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
`Cos`	$\cos(z)$	Cosine
`Pi`	$\pi$	The constant pi (3.14...)
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("93a877"),
    Formula(Equal(Totient(n), Sum(Mul(GCD(n, k), Cos(Div(Mul(Mul(2, Pi), 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