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

$J_k(n)=n^k\prod _{p|n}(1-\frac{1}{p^k})$, 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.

• $\sum _{\delta \mid n}{\delta }^s{J}_r(\delta )J_s\left(\frac{n}{\delta }\right)=J_{r+s}(n)$.

Order of matrix groups

• The general linear group of matrices of order $m$ over $𝐙/n$ has order

$|\mathrm{GL}(m,𝐙/n)|=n^{\frac{m(m-1)}{2}}\prod _{k=1}^m{J}_k(n).$

• The special linear group of matrices of order $m$ over $𝐙/n$ has order

$|\mathrm{SL}(m,𝐙/n)|=n^{\frac{m(m-1)}{2}}\prod _{k=2}^m{J}_k(n).$

• The symplectic group of matrices of order $m$ over $𝐙/n$ has order

$|\mathrm{Sp}(2m,𝐙/n)|=n^{m^2}\prod _{k=1}^m{J}_{2k}(n).$

The first two formulas were discovered by Jordan.

Examples

• Explicit lists in the OEIS are J₂ in OEIS: A007434, J₃ in OEIS: A059376, J₄ in OEIS: A059377, J₅ in OEIS: A059378, J₆ up to J₁₀ in OEIS: A069091 up to OEIS: A069095.
• Multiplicative functions defined by ratios are J₂(n)/J₁(n) in OEIS: A001615, J₃(n)/J₁(n) in OEIS: A160889, J₄(n)/J₁(n) in OEIS: A160891, J₅(n)/J₁(n) in OEIS: A160893, J₆(n)/J₁(n) in OEIS: A160895, J₇(n)/J₁(n) in OEIS: A160897, J₈(n)/J₁(n) in OEIS: A160908, J₉(n)/J₁(n) in OEIS: A160953, J₁₀(n)/J₁(n) in OEIS: A160957, J₁₁(n)/J₁(n) in OEIS: A160960.
• Examples of the ratios J_2k(n)/J_k(n) are J₄(n)/J₂(n) in OEIS: A065958, J₆(n)/J₃(n) in OEIS: A065959, and J₈(n)/J₄(n) in OEIS: A065960.