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Concept
Jordan's totient function
Summary
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 .
Official source
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