Jordan's totient function explained

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 asJ ^{k} _{k(n) = n} \prod _{p|n} \left( 1- (1)/(p ^{k}) \right) , where p ranges through the prime divisors of n .

Properties

\sum _{d} 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

\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} (J _{k(n)})/(n ^{s}) = (\zeta(s - k))/(\zeta(s)) .

• An average order of

J _{k(n)} is J _{k(n)} \sim (n ^{k})/(\zeta(k +1 )) .

• The Dedekind psi function is

\psi(n) = (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 (J _{k(n)})/(J _{1(n)}) or (J _{2k} (n))/(J _{k(n)}) can also be shown to be integer-valued multiplicative functions.\sum _{\delta\mid} \delta ^{sJ} _{r(\delta)J} ^{s\left( (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, Z /n)| = n ^{(m(m -1 ))/(2)} \prod ^{m} _{k =1} J _{k(n).}

• The special linear group of matrices of order

m over Z /n has order|\operatorname{SL}(m, Z /n)| = n ^{(m(m -1 ))/(2)} \prod ^{m} _{k =2} J _{k(n).}

• The symplectic group of matrices of order

m over Z /n has order|\operatorname{Sp}( 2 m, Z /n)| = n ^{m ^{2}} \prod ^{m} _{k =1} 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 J_2k(n)/J_k(n) are J₄(n)/J₂(n) in, J₆(n)/J₃(n) in, and J₈(n)/J₄(n) in .

References

• Book: L. E. Dickson . Leonard Eugene Dickson . . 1919 . 1971 . . 0-8284-0086-5 . 47.0100.04 . 147 .
• Book: Problems in Analytic Number Theory . M. Ram Murty . M. Ram Murty . 206 . . . 2001 . 0-387-95143-1 . 0971.11001 . 11 .
• Book: Sándor . Jozsef . Crstici . Borislav . Handbook of number theory II . Dordrecht . Kluwer Academic . 2004 . 1-4020-2546-7 . 32–36 . 1079.11001 .

External links

• Dorin Andrica . Dorin . Andrica . Mihai . Piticari . On some extensions of Jordan's arithmetic functions. Acta Universitatis Apulensis. 2004. 7. 13–22. 2157944.
• Web site: Matthew . Holden . Michael . Orrison . Michael . Vrable . Yet Another Generalization of Euler's Totient Function . 2011-12-21 . https://web.archive.org/web/20160305132124/https://www.math.hmc.edu/~orrison/research/papers/totient.pdf . 2016-03-05 . dead .

Notes and 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
   1. External links
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