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 asSándor & Crstici (2004) p.106
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.
.Holden et al in external links. The formula is Gegenbauer's.
Order of matrix groups
The general linear group of matrices of order over has orderAll of these formulas are from Andrica and Piticari in #External links.
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
External links