# Jordan's totient function

Let k be a positive integer. In number theory, Jordan's totient function $J_k(n)$ of a positive integer n is 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. The function is named after Camille Jordan.

## Definition

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

$J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) .\,$

## Properties

• $\sum_{d | n } J_k(d) = n^k. \,$

which may be written in the language of Dirichlet convolutions as[1]

$J_k(n) \star 1 = n^k\,$

and via Möbius inversion as

$J_k(n) = \mu(n) \star n^k$.

Since the Dirichlet generating function of μ is 1/ζ(s) and the Dirichlet generating function of nk is ζ(s-k), the series for Jk becomes

$\sum_{n\ge 1}\frac{J_k(n)}{n^s} = \frac{\zeta(s-k)}{\zeta(s)}$.
$\frac{n^k}{\zeta(k+1)}$.
$\psi(n) = \frac{J_2(n)}{J_1(n)}$,

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.

• $\sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)$.      [2]

## Order of matrix groups

The general linear group of matrices of order m over Zn has order[3]

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

The special linear group of matrices of order m over Zn 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 Zn 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 .

## Notes

1. ^ Sándor & Crstici (2004) p.106
2. ^ Holden et al in external links The formula is Gegenbauer's
3. ^ All of these formulas are from Andrici and Priticari in #External links