On the image of Euler’s totient function

Euler's totient function φ is the function defined on the positive natural numbers N * in the following way: if n ∈ N * , then φ(n) is the cardinal of the set {x ∈ N * : 1 ≤ x ≤ n, (x, n) = 1}, where (x, n) is the pgcd of x and n. Thus φ(1) = 1, φ(2) = 1, φ(3) = 2, φ(4) = 2, and so on. The principle aim of this article is to study certain aspects of the image of the function φ. 1 Elementary properties Clearly φ(p) = p − 1, for any prime number p and, more generally, if α ∈ N * , then φ(p α) = p α − p α−1. This follows from the fact that the only numbers which are not coprime with p α are multiples of p and there are p α−1 such multiples x with 1 ≤ x ≤ p α .