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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In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form It can be resumed formally by expanding the denominator: where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1: This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform. Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series.
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.
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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Springer International Publishing2017
Explores commutative groups, Euler's Totient Function, and Cartesian products in group theory.
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Explores the Chinese remainder theorem, systems of congruences, and Euclidean domains in integer numbers and polynomial rings.
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