An asymptotic formula for the number of totients

Let $V(x)$ count the distinct values of Euler's totient function up to x. We give an explicit asymptotic equivalent for $V(x)$. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that $V(cx)/V(x)\to c$ as $x\to \infty$ for every fixed c > 0, answering a question of Erdős and Hall.