I don't know if there is any done work done about ehis matter, and I don't have access to research news.
I'm interested in this question (I haen't tried to answer it myself, but it seems very difficult to solve with a pen and a notebook at home :-) ):
Let be $p_n$ the $n$th prime. Is the sequence $\left\{\frac{p_{n+1}-p_n}n\right\}$ bounded?
I know the Prime Number Theorem, but I'm not sure if this question can be answered with it.
It's known that $p_{n+1}-p_n<n^c$ for some constant $c$, $1/2<c<1$.
See the "upper bounds" section of http://en.wikipedia.org/wiki/Prime_gap