# Jordan totient function

From Encyclopedia of Mathematics

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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, 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

**How to Cite This Entry:**

Jordan totient function.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Jordan_totient_function&oldid=34695