Jordan's totient function

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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 J1. The function is named after Camille Jordan.

Definition[edit]

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[edit]

  • \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 μ is 1/ζ(s) and the Dirichlet generating function of nk is ζ(s-k), the series for Jk becomes

\sum_{n\ge 1}\frac{J_k(n)}{n^s} = \frac{\zeta(s-k)}{\zeta(s)}.
\frac{n^k}{\zeta(k+1)}.
\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^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)
.      [2]

Order of matrix groups[edit]

The general linear group of matrices of order m over Zn 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 Zn 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 Zn 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[edit]

Explicit lists in the OEIS are J2 in OEISA007434, J3 in OEISA059376, J4 in OEISA059377, J5 in OEISA059378, J6 up to J10 in OEISA069091 up to OEISA069095.


Multiplicative functions defined by ratios are J2(n)/J1(n) in OEISA001615, J3(n)/J1(n) in OEISA160889, J4(n)/J1(n) in OEISA160891, J5(n)/J1(n) in OEISA160893, J6(n)/J1(n) in OEISA160895, J7(n)/J1(n) in OEISA160897, J8(n)/J1(n) in OEISA160908, J9(n)/J1(n) in OEISA160953, J10(n)/J1(n) in OEISA160957, J11(n)/J1(n) in OEISA160960.


Examples of the ratios J2k(n)/Jk(n) are J4(n)/J2(n) in OEISA065958, J6(n)/J3(n) in OEISA065959, and J8(n)/J4(n) in OEISA065960.

Notes[edit]

  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[edit]

External links[edit]