Jordan's totient function

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

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]].