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
:$J_k(n)=n^k \prod_\left(1-\frac\right) \,$, where $p$ ranges through the prime divisors of $n$.

Properties

* $\sum_ J_k(d) = n^k. \,$
: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
::$\sum_\frac = \frac$.

* An average order of $J_k(n)$ is
::$J_k(n) \sim \frac$.

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

* $\sum_\delta^sJ_r(\delta)J_s\left(\frac\right) = J_(n)$.

Order of matrix groups

* The general linear group of matrices of order $m$ over $\mathbf/n$ has orderAll of these formulas are from Andrica and Piticari in #External links.
:$, \operatorname(m,\mathbf/n), =n^\prod_^m J_k(n).$

* The special linear group of matrices of order $m$ over $\mathbf/n$ has order
:$, \operatorname(m,\mathbf/n), =n^\prod_^m J_k(n).$

* The symplectic group of matrices of order $m$ over $\mathbf/n$ has order
:$, \operatorname(2m,\mathbf/n), =n^\prod_^m J_(n).$

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 J_2k(n)/J_k(n) are J₄(n)/J₂(n) in , J₆(n)/J₃(n) in , and J₈(n)/J₄(n) in .

Notes

References

*
*
*

External links

*
*

{{Totient

Modular arithmetic
Multiplicative functions