Jordan's totient function

Let $k$ be a positive integer. In number theory, Jordan's totient function $J_{k}(n)$ of a positive integer $n$ 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 a generalization of Euler's totient function, which is given by $J_{1}(n)$ . The function is named after Camille Jordan.

Definition

For each $k$ , 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
$\sum _{n\geq 1}{\frac {J_{k}(n)}{n^{s}}}={\frac {\zeta (s-k)}{\zeta (s)}}$ .
• An average order of $J_{k}(n)$ is
${\frac {n^{k}}{\zeta (k+1)}}$ .
$\psi (n)={\frac {J_{2}(n)}{J_{1}(n)}}$ ,
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.
• $\sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)$ .      

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 .