Euler phi function

For any positive integer n, φ(n) is the number of positive integers less than or equal to n which are coprimeMathworldPlanetmath to n. The function φ is known as the φ function. This function may also be denoted by ϕ.

Among the most useful of φ are the facts that it is multiplicative (meaning if gcd(a,b)=1, then φ(ab)=φ(a)φ(b)) and that φ(pk)=pk-1(p-1) for any prime p and any positive integer k. These two facts combined give a numeric computation of φ for all positive integers:

φ(n) =np|n(1-1p). (1)

For example,

φ(2000) =2000p|2000(1-1p)

From equation (1), it is clear that φ(n)n for any positive integer n with equality holding exactly when n=1. This is because


with equality holding exactly when n=1.

Another important fact about the φ function is that


where the sum extends over all positive divisorsMathworldPlanetmathPlanetmathPlanetmath of n. Also, by definition, φ(n) is the number of units in the ring /n of integers modulo n.

The sequence {φ(n)} appears in the OEIS as sequence njas/sequences/A000010A000010.

Title Euler phi function
Canonical name EulerPhiFunction
Date of creation 2013-03-22 11:45:11
Last modified on 2013-03-22 11:45:11
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 19
Author Wkbj79 (1863)
Entry type Definition
Classification msc 11A25
Classification msc 11-00
Classification msc 14F35
Classification msc 14H30
Classification msc 20F34
Synonym Euler totient function
Synonym Euler φ function
Synonym Euler ϕ function
Related topic ProofThatEulerPhiIsAMultiplicativeFunction
Related topic ValuesOfNForWhichVarphintaun
Related topic PrimeResidueClass
Related topic EulerPhiAtAProduct
Related topic SummatoryFunctionOfArithmeticFunction