Euler's phi function
Euler's totient function φ(n), named after Leonhard Euler, is an important function in number theory.
If n is a positive integer, then φ(n) is defined to be the number of positive integers less than or equal to n and coprime to n.
For example, φ(8) = 4 since the four numbers 1, 3, 5 and 7 are coprime to 8.
φ is a (conditionally) multiplicative function: if m and n are coprime then φ(mn) = φ(m) φ(n). (Sketch of proof: let
A, B, C be the sets of residue classes
modulo-and-coprime-to m, n, mn respectively; then there is a bijection between AxB and C, via the Chinese Remainder Theorem.)
The value of φ(n) can thus be computed using the fundamental theorem of arithmetic: if n = p1k1 ... prkr
where the pj are distinct primes,
then φ(n) = (p1-1) p1k1-1 ... (pr-1) prkr-1.
(Sketch of proof: the case r = 1 is easy, and the general result follows by multiplicativity.)
The value of φ(n) is equal to the order of the group of units of the ring Z/nZ (see modular arithmetic). This, together with Lagrange's theorem, provides a proof for Euler's theorem.
φ(n) is also equal to the number of generators of the cyclic group Cn (and therefore also to the degree of the cyclotomic polynomial Φn). Since every element of Cn generates a cyclic subgroup and the subgroups of Cn are of the form Cd where d divides n (written as d|n), we get
where the sum extends over all positive divisors d of n.
We can now use the Möbius inversion formula to "invert" this sum and get another formula for φ(n):
A Dirichlet series involving φ(n) is
Referenced By
Blum Blum Shub | Elementary function | Elementary functions | Euler's Theorem | Extended Riemann hypothesis | Fermat's Little Theorem | Fermat-Euler theorem | Fermats little theorem | Fermats little theorom | Generalised Riemann hypothesis | Generalized Riemann hypothesis | List of functions | List of mathematical functions | Number Theory | Primitive root modulo n | Root of unity | Special function | Special functions | Theory of numbers
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