Let
Contents
Definition
Jordan's totient function is multiplicative and may be evaluated as
Properties
which may be written in the language of Dirichlet convolutions as
and via Möbius inversion as
Since the Dirichlet generating function of
and by inspection of the definition (recognizing that each factor in the product over the primes is a cyclotomic polynomial of
Order of matrix groups
The general linear group of matrices of order
The special linear group of matrices of order
The symplectic group of matrices of order
The first two formulas were discovered by Jordan.
Examples
Explicit lists in the OEIS are J2 in A007434, J3 in A059376, J4 in A059377, J5 in A059378, J6 up to J10 in A069091 up to A069095.
Multiplicative functions defined by ratios are J2(n)/J1(n) in A001615, J3(n)/J1(n) in A160889, J4(n)/J1(n) in A160891, J5(n)/J1(n) in A160893, J6(n)/J1(n) in A160895, J7(n)/J1(n) in A160897, J8(n)/J1(n) in A160908, J9(n)/J1(n) in A160953, J10(n)/J1(n) in A160957, J11(n)/J1(n) in A160960.
Examples of the ratios J2k(n)/Jk(n) are J4(n)/J2(n) in A065958, J6(n)/J3(n) in A065959, and J8(n)/J4(n) in A065960.