Jordan's totient function

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

$J_{k}(n)\star 1=n^{k}\,$

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

$\sum _{n\geq 1}{\frac {J_{k}(n)}{n^{s}}}={\frac {\zeta (s-k)}{\zeta (s)}}$.

• The average order of J_k(n) is c n^k for some c.

• 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.

• $\sum _{\delta \mid n}\delta ^{s}J_{r}(\delta )J_{s}\left({\frac {n}{\delta }}\right)=J_{r+s}(n)$      

Order of matrix groups

The general linear group of matrices of order m over Z_n has order

$|GL(m,Z_{n})|=n^{\frac {m(m-1)}{2}}\prod _{k=1}^{m}J_{k}(n).$

The special linear group of matrices of order m over Z_n has order

$|SL(m,Z_{n})|=n^{\frac {m(m-1)}{2}}\prod _{k=2}^{m}J_{k}(n).$

The symplectic group of matrices of order m over Z_n has order

$|Sp(2m,Z_{n})|=n^{m^{2}}\prod _{k=1}^{m}J_{2k}(n).$

The first two formulas were discovered by Jordan.

References

• L. E. Dickson (1919, repr.1971). History of the Theory of Numbers I. Chelsea. p. 147. ISBN 0-8284-0086-5. {{cite book}}: Check date values in: |year= (help)CS1 maint: year (link)
• M. Ram Murty (2001). Problems in Analytic Number Theory. Graduate Texts in Mathematics. Vol. 206. Springer-Verlag. p. 11. ISBN 0-387-95143-1.

External links

Dorin Andrica and Mihai Piticari On some Extensions of Jordan's arithmetical Functions

Matthew Holden, Michael Orrison, Michael Varble Yet another Generalization of Euler's Totient Function