Jordan totient \(J_k(n)\) refresher

Jordan’s totient function generalizes Euler’s totient. It is multiplicative and appears naturally in group-counting problems. See [Apostol, 1976] and [Tenenbaum, 2015].

Definition

For a fixed integer \(k\ge 1\), the Jordan totient function \(J_k(n)\) can be defined by

\[ J_k(n)=n^k \prod_{p\mid n}\left(1-\frac{1}{p^k}\right). \]

For \(k=1\), this is Euler’s totient: \(J_1(n)=\varphi(n)\).

Divisor-sum identity

A useful identity is

\[ \sum_{d\mid n} J_k(d)=n^k. \]

In Dirichlet convolution language:

\[ J_k \ast 1 = \mathrm{id}^k, \]

hence \(J_k = \mu \ast \mathrm{id}^k\).

Experiment ideas

• compare \(J_k(n)\) as \(k\) varies

• visualize \(J_2(n)\) and \(J_3(n)\) over \(n\le N\)

• compare \(J_k(n)/n^k\) as a “prime factor penalty”

Experiments in this repository

• E099 — Jordan totients J_k: atlas, identities (J_1=φ), and scaling J_k(n)/n^k.