Jordan totient function

2020 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, ed. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp. 32–36. ISBN 1-4020-2546-7 Zbl 1079.11001