In number theory

Number theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

, Jordan's totient function of a positive integer n is the number of k-tuples of positive integers all less than or equal to n that form a coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 (k + 1)-tuple together with n. This is a generalisation of Euler's totient function, which is J₁. The function is named after Camille Jordan

Camille Jordan

Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...

.

Definition

Jordan's totient function is multiplicative

Multiplicative function

In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...

 and may be evaluated as

Properties

• 

which may be written in the language of Dirichlet convolution

Dirichlet convolution

In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Johann Peter Gustav Lejeune Dirichlet, a German mathematician.-Definition:...

s as

and via Möbius inversion

Möbius inversion formula

In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. Other Möbius inversion formulas are obtained when different local finite partially ordered sets replace the classic case of the natural numbers ordered by...

 as.
Since the Dirichlet generating function of μ is 1/ζ(s) and the
Dirichlet generating function of n^k is ζ(s-k), the series for
J_k becomes.

• The average order
  Average order of an arithmetic function
  In number theory, the average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".Let f be an arithmetic function...
  
   of J_k(n) is c n^k for some c.
• 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 p^-k), the arithmetic functions
defined by or
can also be shown to be integer-valued multiplicative functions.

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 J_2k(n)/J_k(n) are
J₄(n)/J₂(n) in ,
J₆(n)/J₃(n) in ,
and
J₈(n)/J₄(n) in .