# Jordan totient function

2010 Mathematics Subject Classification: Primary: 11A [MSN][ZBL]

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 $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's totient function, which is $J_1$.

Jordan's totient function is multiplicative 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 $J_k(n)$ is $c n^k$ for some $c$.

## References

• Dickson, L.E. History of the Theory of Numbers I, Chelsea (1971) p. 147, Template:ISBN
• Ram Murty, M. Problems in Analytic Number Theory, ser. Graduate Texts in Mathematics 206 Springer-Verlag (2001) p. 11. Template:ISBN Zbl 0971.11001
• Sándor, Jozsef; Crstici, Borislav, ed. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp. 32–36. Template:ISBN Zbl 1079.11001
How to Cite This Entry:
Jordan totient function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_totient_function&oldid=52987