Definition:Euler Phi Function

Definition
Let $n \in \Z_{>0}$, that is, a strictly positive integer.

The Euler $\phi$ (phi) function is the function $\phi: \Z_{>0} \to \Z_{>0}$ defined as:


 * $\phi \left({n}\right) = $ the number of integers less than or equal to $n$ which are prime to $n$

That is:
 * $\phi \left({n}\right) = \left|{S_n}\right|: S_n = \left\{{k: 1 \le k \le n, k \perp n}\right\}$

Note that by this definition $\phi \left({1}\right) = 1$ as $\gcd \left\{{1, 1}\right\} = 1$.

It follows from the definition of $\Z'_n$ that $\phi \left({n}\right)$ is the number of elements in $\Z'_n$.

Also known as
The Euler $\phi$ function is also known as the totient function or the indicator function

Some sources render it with a hyphen: Euler $\phi$-function.