# Jordan's totient function

Let be a positive integer. In number theory, Jordan's totient function of a positive integer is the number of -tuples of positive integers all less than or equal to that form a coprime -tuple together with . This is a generalisation of Euler's totient function, which is . The function is named after Camille Jordan.

## Definition

Jordan's totient function is multiplicative and may be evaluated as

## Properties

which may be written in the language of Dirichlet convolutions as[1]

and via Möbius inversion as

.

Since the Dirichlet generating function of is and the Dirichlet generating function of is , the series for becomes

.
.
,

and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of ), the arithmetic functions defined by or can also be shown to be integer-valued multiplicative functions.

## Order of matrix groups

The general linear group of matrices of order over has order[3]

The special linear group of matrices of order over has order

The symplectic group of matrices of order over has order

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

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