Jordan's totient function

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

Definition

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

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

Properties

• $\sum _{{d|n}}J_{k}(d)=n^{k}.\,$

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

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

and via Möbius inversion as

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

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

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

• 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 ${\displaystyle p_{-k}}$), 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.

• $\sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)$.      ^[2]

Order of matrix groups

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

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

The special linear group of matrices of order $m$ over ${\displaystyle \mathbf {Z} _{n}}$ 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 ${\displaystyle \mathbf {Z} _{n}}$ 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  A007434, J₃ in  A059376, J₄ in  A059377, J₅ in  A059378, J₆ up to J₁₀ in  A069091 up to  A069095.

Multiplicative functions defined by ratios are J₂(n)/J₁(n) in  A001615, J₃(n)/J₁(n) in  A160889, J₄(n)/J₁(n) in  A160891, J₅(n)/J₁(n) in  A160893, J₆(n)/J₁(n) in  A160895, J₇(n)/J₁(n) in  A160897, J₈(n)/J₁(n) in  A160908, J₉(n)/J₁(n) in  A160953, J₁₀(n)/J₁(n) in  A160957, J₁₁(n)/J₁(n) in  A160960.

Examples of the ratios J_2k(n)/J_k(n) are J₄(n)/J₂(n) in  A065958, J₆(n)/J₃(n) in  A065959, and J₈(n)/J₄(n) in  A065960.

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

References

• L. E. Dickson (1971) [1919]. History of the Theory of Numbers, Vol. I. Chelsea Publishing. p. 147. ISBN 0-8284-0086-5. JFM 47.0100.04.
• M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. 206. Springer-Verlag. p. 11. ISBN 0-387-95143-1. Zbl 0971.11001.
• Sándor, Jozsef; Crstici, Borislav (2004). Handbook of number theory II. Dordrecht: Kluwer Academic. pp. 32–36. ISBN 1-4020-2546-7. Zbl 1079.11001.

External links

• Andrica, Dorin; Piticari, Mihai (2004). "On some Extensions of Jordan's arithmetical Functions" (PDF). Acta universitatis Apulensis (7). MR 2157944.
• Holden, Matthew; Orrison, Michael; Varble, Michael. "Yet another Generalization of Euler's Totient Function" (PDF).