Jordan's totient function
In number theory, Jordan's totient function, denoted as , where  is a positive integer, is a function of a positive integer, , that equals the number of -tuples of positive integers that are less than or equal to  and that together with  form a coprime set of  integers 

Jordan's totient function is a generalization of Euler's totient function, which is the same as . The function is named after Camille Jordan. Definition

For each positive integer , Jordan's totient function  is multiplicative and may be evaluated as
, where  ranges through the prime divisors of . Properties

 
which may be written in the language of Dirichlet convolutions as
 
and via Möbius inversion as
. Since the Dirichlet generating function of  is  and the Dirichlet generating function of  is , the series for  becomes
. An average order of  is  
. The Dedekind psi function is
,
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

 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

References

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Modular arithmetic
Multiplicative functions