Jordan's totient function

In number theory, Jordan's totient function, denoted as ${\displaystyle J_{k}(n)}$, where ${\displaystyle k}$ is a positive integer, is a function of a positive integer, ${\displaystyle n}$, that equals the number of ${\displaystyle k}$-tuples of positive integers that are less than or equal to ${\displaystyle n}$ and that together with ${\displaystyle n}$ form a coprime set of ${\displaystyle k+1}$ integers.

Jordan's totient function is a generalization of Euler's totient function, which is the same as ${\displaystyle J_{1}(n)}$. The function is named after Camille Jordan.

Definition

For each positive integer ${\displaystyle k}$, Jordan's totient function ${\displaystyle J_{k}}$ is multiplicative and may be evaluated as

${\displaystyle J_{k}(n)=n^{k}\prod _{p|n}\left(1-{\frac {1}{p^{k}}}\right)\,}$, where ${\displaystyle p}$ ranges through the prime divisors of ${\displaystyle n}$.

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

${\displaystyle J_{k}(n)\sim {\frac {n^{k}}{\zeta (k+1)}}}$.

• The Dedekind psi function is

${\displaystyle \psi (n)={\frac {J_{2}(n)}{J_{1}(n)}}}$,

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 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.