For a prime number $p$, let $\Phi(p)$ be the subset of $\{ 1, 2, \ldots, p-1 \}$ consisting of primitive roots modulo $p$. (Thus $\# \Phi(p) = \phi(p-1)$, where $\phi$ denotes the totient.)

I am curious about the distribution of $\Phi(p)$ as $p$ varies. I imagine that this is classic analytic number theory, but I'm finding it surprisingly hard to locate a precise statement/reference.

To make my question precise, consider the following: For any continuous function $f \colon [0,1] \rightarrow {\mathbb R}$, define $$D_p(f) = \frac{1}{\phi(p-1)} \cdot \sum_{x \in \Phi(p)} f \left( \frac{x}{p-1} \right).$$

Does this sequence of distributions $D_p$ converge (e.g., weakly in measure, or in some stronger sense?) to the uniform measure or something else on $[0,1]$? What's the best current result along these lines?