Jordan's totient function

Short description: A function in mathematics, number theory

In number theory, Jordan's totient function, denoted as [math]\displaystyle{ J_k(n) }[/math], where [math]\displaystyle{ k }[/math] is a positive integer, is a function of a positive integer, [math]\displaystyle{ n }[/math], that equals the number of [math]\displaystyle{ k }[/math]-tuples of positive integers that are less than or equal to [math]\displaystyle{ n }[/math] and that together with [math]\displaystyle{ n }[/math] form a coprime set of [math]\displaystyle{ k+1 }[/math] integers

Jordan's totient function is a generalization of Euler's totient function, which is the same as [math]\displaystyle{ J_1(n) }[/math]. The function is named after Camille Jordan.

Definition

For each positive integer [math]\displaystyle{ k }[/math], Jordan's totient function [math]\displaystyle{ J_k }[/math] is multiplicative and may be evaluated as

[math]\displaystyle{ J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) \, }[/math], where [math]\displaystyle{ p }[/math] ranges through the prime divisors of [math]\displaystyle{ n }[/math].

Properties

• [math]\displaystyle{ \sum_{d | n } J_k(d) = n^k. \, }[/math]

which may be written in the language of Dirichlet convolutions as

[math]\displaystyle{ J_k(n) \star 1 = n^k\, }[/math]

and via Möbius inversion as

[math]\displaystyle{ J_k(n) = \mu(n) \star n^k }[/math].

Since the Dirichlet generating function of [math]\displaystyle{ \mu }[/math] is [math]\displaystyle{ 1/\zeta(s) }[/math] and the Dirichlet generating function of [math]\displaystyle{ n^k }[/math] is [math]\displaystyle{ \zeta(s-k) }[/math], the series for [math]\displaystyle{ J_k }[/math] becomes

[math]\displaystyle{ \sum_{n\ge 1}\frac{J_k(n)}{n^s} = \frac{\zeta(s-k)}{\zeta(s)} }[/math].

• An average order of [math]\displaystyle{ J_k(n) }[/math] is

[math]\displaystyle{ J_k(n) \sim \frac{n^k}{\zeta(k+1)} }[/math].

• The Dedekind psi function is

[math]\displaystyle{ \psi(n) = \frac{J_2(n)}{J_1(n)} }[/math],

and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of [math]\displaystyle{ p^{-k} }[/math]), the arithmetic functions defined by [math]\displaystyle{ \frac{J_k(n)}{J_1(n)} }[/math] or [math]\displaystyle{ \frac{J_{2k}(n)}{J_k(n)} }[/math] can also be shown to be integer-valued multiplicative functions.

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

Order of matrix groups

• The general linear group of matrices of order [math]\displaystyle{ m }[/math] over [math]\displaystyle{ \mathbf{Z}/n }[/math] has order

[math]\displaystyle{ |\operatorname{GL}(m,\mathbf{Z}/n)|=n^{\frac{m(m-1)}{2}}\prod_{k=1}^m J_k(n). }[/math]

• The special linear group of matrices of order [math]\displaystyle{ m }[/math] over [math]\displaystyle{ \mathbf{Z}/n }[/math] has order

[math]\displaystyle{ |\operatorname{SL}(m,\mathbf{Z}/n)|=n^{\frac{m(m-1)}{2}}\prod_{k=2}^m J_k(n). }[/math]

• The symplectic group of matrices of order [math]\displaystyle{ m }[/math] over [math]\displaystyle{ \mathbf{Z}/n }[/math] has order

[math]\displaystyle{ |\operatorname{Sp}(2m,\mathbf{Z}/n)|=n^{m^2}\prod_{k=1}^m J_{2k}(n). }[/math]

The first two formulas were discovered by Jordan.

Examples

• Explicit lists in the OEIS are J₂ in OEIS: A007434, J₃ in OEIS: A059376, J₄ in OEIS: A059377, J₅ in OEIS: A059378, J₆ up to J₁₀ in OEIS: A069091 up to OEIS: A069095.

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

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

References

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

External links

• Andrica, Dorin; Piticari, Mihai (2004). "On some extensions of Jordan's arithmetic functions". Acta Universitatis Apulensis 7: 13-22. https://eudml.org/doc/126410. 
• Holden, Matthew; Orrison, Michael; Vrable, Michael. "Yet Another Generalization of Euler's Totient Function". http://www.math.hmc.edu/~orrison/research/papers/totient.pdf.