# Jordan's totient function

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In number theory, Jordan's totient function ${\displaystyle J_{k}(n)}$ of a positive integer n, named after Camille Jordan, is defined to be the number of k-tuples of positive integers all less than or equal to n that form a coprime (k + 1)-tuple together with n. This is a generalisation of Euler's totient function, which is J1.

## Definition

Jordan's totient function is multiplicative and may be evaluated as

${\displaystyle J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right).\,}$

## Properties

• ${\displaystyle \sum _{d|n}J_{k}(d)=n^{k}\,}$.
• The average order of Jk(n) is c nk for some c.