Jordan's totient function

Let be a positive integer. In number theory, the Jordan's totient function of a positive integer 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 given by . The function is named after Camille Jordan. For each , Jordan's totient function is multiplicative and may be evaluated as where ranges through the prime divisors of . 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. 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. 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 .

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Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8.

Möbius inversion formula

In mathematics, the classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced into number theory in 1832 by August Ferdinand Möbius. A large generalization of this formula applies to summation over an arbitrary locally finite partially ordered set, with Möbius' classical formula applying to the set of the natural numbers ordered by divisibility: see incidence algebra.

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