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

For each, 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 is and the
Dirichlet generating function of is, the series for
becomes
and by inspection of the definition, the arithmetic functions
defined by or
can also be shown to be integer-valued multiplicative functions.
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/J1 in,
J3/J1 in,
J4/J1 in,
J5/J1 in,
J6/J1 in,
J7/J1 in,
J8/J1 in,
J9/J1 in,
J10/J1 in,
J11/J1 in.

Examples of the ratios J2k/Jk are
J4/J2 in,
J6/J3 in,
and J8/J4 in.