Jordan's totient function

In number theory, Jordan's totient function, denoted as $$J_k(n)$$, where $$k$$ is a positive integer, is a function of a positive integer, $$n$$, that equals the number of $$k$$-tuples of positive integers that are less than or equal to $$n$$ and that together with $$n$$ form a coprime set of $$k+1$$ integers

Jordan's totient function is a generalization of Euler's totient function, which is the same as $$J_1(n)$$. The function is named after Camille Jordan.

Definition
For each positive integer $$k$$, Jordan's totient function $$J_k$$ is multiplicative and may be evaluated as
 * $$J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) \,$$, where $$p$$ ranges through the prime divisors of $$n$$.

Properties

 * $$\sum_{d | n } J_k(d) = n^k. \, $$
 * which may be written in the language of Dirichlet convolutions as
 * $$J_k(n) \star 1 = n^k\,$$
 * and via Möbius inversion as
 * $$J_k(n) = \mu(n) \star n^k$$.
 * Since the Dirichlet generating function of $$\mu$$ is $$1/\zeta(s)$$ and the Dirichlet generating function of $$n^k$$ is $$\zeta(s-k)$$, the series for $$J_k$$ becomes
 * $$\sum_{n\ge 1}\frac{J_k(n)}{n^s} = \frac{\zeta(s-k)}{\zeta(s)}$$.


 * An average order of $$J_k(n)$$ is
 * $$ J_k(n) \sim \frac{n^k}{\zeta(k+1)}$$.


 * The Dedekind psi function is
 * $$\psi(n) = \frac{J_2(n)}{J_1(n)}$$,
 * and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of $$p^{-k}$$), the arithmetic functions defined by $$\frac{J_k(n)}{J_1(n)}$$ or $$\frac{J_{2k}(n)}{J_k(n)}$$ can also be shown to be integer-valued multiplicative functions.


 * $$\sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)$$.

Order of matrix groups

 * The general linear group of matrices of order $$m$$ over $$\mathbf{Z}/n$$ has order

$$
 * \operatorname{GL}(m,\mathbf{Z}/n)|=n^{\frac{m(m-1)}{2}}\prod_{k=1}^m J_k(n).


 * The special linear group of matrices of order $$m$$ over $$\mathbf{Z}/n$$ has order

$$
 * \operatorname{SL}(m,\mathbf{Z}/n)|=n^{\frac{m(m-1)}{2}}\prod_{k=2}^m J_k(n).


 * The symplectic group of matrices of order $$m$$ over $$\mathbf{Z}/n$$ has order

$$
 * \operatorname{Sp}(2m,\mathbf{Z}/n)|=n^{m^2}\prod_{k=1}^m J_{2k}(n).

The first two formulas were discovered by Jordan.

Examples

 * Explicit lists in the OEIS are J2 in, J3 in , J4 in , J5 in , J6 up to J10 in up to.
 * Multiplicative functions defined by ratios are J2(n)/J1(n) in, J3(n)/J1(n) in , J4(n)/J1(n) in , J5(n)/J1(n) in , J6(n)/J1(n) in , J7(n)/J1(n) in , J8(n)/J1(n) in , J9(n)/J1(n) in , J10(n)/J1(n) in , J11(n)/J1(n) in.
 * Examples of the ratios J2k(n)/Jk(n) are J4(n)/J2(n) in, J6(n)/J3(n) in , and J8(n)/J4(n) in.