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}\left(1-\frac{1}{p^k}\right) ,, 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
: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^sJ_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 \mathbf{Z}/n has order
:
|\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 \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 \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 , 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 J2k(n)/Jk(n) are J4(n)/J2(n) in , J6(n)/J3(n) in , and J8(n)/J4(n) in .
## Notes
## References
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## References
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 Andrica and Piticari in [[#External links]].
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