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 asProperties
- An average order of is
- The Dedekind psi function is
- .
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
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.