**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 *A*x*B* and *C*, via the Chinese Remainder Theorem.)

The value of φ(*n*) can thus be computed using the fundamental theorem of arithmetic: if *n* = *p*_{1}^{k1} ... *p*_{r}^{kr}
where the *p*_{j} are distinct primes,
then φ(*n*) = (*p*_{1}-1) *p*_{1}^{k1-1} ... (*p*_{r}-1) *p*_{r}^{kr-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**/*n***Z** (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 *C*_{n} (and therefore also to the degree of the cyclotomic polynomial Φ_{n}). Since every element of *C*_{n} generates a cyclic subgroup and the subgroups of *C*_{n} are of the form *C*_{d} where *d* divides *n* (written as *d*|*n*), we get

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