SOLVED
Let $N_k=2\cdot 3\cdots p_k$ and $\{a_1<a_2<\cdots <a_{\phi(N_k)}\}$ be the integers $<N_k$ which are relatively prime to $N_k$. Then, for any $c\geq 0$, the limit
\[\frac{\#\{ a_i-a_{i-1}\leq c \frac{N_k}{\phi(N_k)} : 2\leq i\leq \phi(N_k)\}}{\phi(N_k)}\]
exists and is a continuous function of $c$.
Solved by Hooley
[Ho65], who proved that these gaps have an exponential distribution: that is, if $f(c)$ is the function in question, then
\[f(c)=(1+o(1))(1-e^{-c})\]
(where the $o(1)$ goes to $0$ uniformly as $k\to \infty$).
