View source for Jordan totient function

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{{MSC|11A}}

An arithmetic function $J_k(n)$ of a [[natural number]] $n$, named after [[Camille Jordan]], counting the $k$-tuples of positive integers all less than or equal to $n$ that form a [[Coprime numbers|coprime]] $(k + 1)$-tuple together with $n$.  This is a generalisation of Euler's [[totient function]], which is $J_1$.

Jordan's totient function is [[Multiplicative arithmetic function|multiplicative]] and may be evaluated as
$$
J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) \ .
$$

By [[Möbius inversion]] we have $\sum_{d | n } J_k(d) = n^k $.  The [[Average order of an arithmetic function|average order]] of $J_k(n)$ is $c n^k$ for some $c$.

==References==
* Dickson, L.E. History of the Theory of Numbers I, Chelsea (1971) p. 147,  ISBN 0-8284-0086-5
* Ram Murty, M. Problems in Analytic Number Theory, ser. Graduate Texts in Mathematics '''206''' Springer-Verlag (2001) p. 11  ISBN 0387951431 {{ZBL|0971.11001}}
* Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.32–36. ISBN 1-4020-2546-7. {{ZBL|1079.11001}}