Jordan's totient function

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Let ${\displaystyle k}$ be a positive integer. In number theory, Jordan's totient function ${\displaystyle J_{k}(n)}$ of a positive integer ${\displaystyle n}$ is the number of ${\displaystyle k}$-tuples of positive integers all less than or equal to ${\displaystyle n}$ that form a coprime ${\displaystyle (k+1)}$-tuple together with ${\displaystyle n}$. This is a generalisation of Euler's totient function, which is ${\displaystyle J_{1}}$. The function is named after Camille Jordan.

Definition

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

${\displaystyle J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right).\,}$

Properties

• ${\displaystyle \sum _{d|n}J_{k}(d)=n^{k}.\,}$

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

${\displaystyle J_{k}(n)\star 1=n^{k}\,}$

and via Möbius inversion as

${\displaystyle J_{k}(n)=\mu (n)\star n^{k}}$.

Since the Dirichlet generating function of ${\displaystyle \mu }$ is ${\displaystyle 1/\zeta (s)}$ and the Dirichlet generating function of ${\displaystyle n^{k}}$ is ${\displaystyle \zeta (s-k)}$, the series for ${\displaystyle J_{k}}$ becomes

${\displaystyle \sum _{n\geq 1}{\frac {J_{k}(n)}{n^{s}}}={\frac {\zeta (s-k)}{\zeta (s)}}}$.
• An average order of ${\displaystyle J_{k}(n)}$ is
${\displaystyle {\frac {n^{k}}{\zeta (k+1)}}}$.
${\displaystyle \psi (n)={\frac {J_{2}(n)}{J_{1}(n)}}}$,

and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of ${\displaystyle p_{-k}}$), the arithmetic functions defined by ${\displaystyle {\frac {J_{k}(n)}{J_{1}(n)}}}$ or ${\displaystyle {\frac {J_{2k}(n)}{J_{k}(n)}}}$ can also be shown to be integer-valued multiplicative functions.

• ${\displaystyle \sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)}$.      [2]

Order of matrix groups

The general linear group of matrices of order ${\displaystyle m}$ over ${\displaystyle \mathbf {Z} _{n}}$ has order[3]

${\displaystyle |\operatorname {GL} (m,\mathbf {Z} _{n})|=n^{\frac {m(m-1)}{2}}\prod _{k=1}^{m}J_{k}(n).}$

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

${\displaystyle |\operatorname {SL} (m,\mathbf {Z} _{n})|=n^{\frac {m(m-1)}{2}}\prod _{k=2}^{m}J_{k}(n).}$

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

${\displaystyle |\operatorname {Sp} (2m,\mathbf {Z} _{n})|=n^{m^{2}}\prod _{k=1}^{m}J_{2k}(n).}$

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