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 J₁. 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 μ is 1/ζ(s) and the Dirichlet generating function of n^k is ζ(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

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

• 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.

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

Order of matrix groups

The general linear group of matrices of order m over 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 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 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).