Zhang's celebrated result established the existence of bounded gaps between primes, that is, there exists a constant $B$ (in Zhang's paper $B$ can be taken to be $7 \times 10^7$, and this was almost immediately improved significantly so that one may take $B= 4680$) such that there are infinitely many primes $p_n$ such that $p_n - p_{n-1} < B$.

My question considers the opposite case. Do there exist, infinitely often, gaps that are much larger than average? The prime number theorem implies that the average gap between $p_n$ and $p_{n+1}$ is about $\log p_n$.

Thus, for which constant $C > 1$ can one establish the existence of infinitely many primes $p_n$ such that $p_n - p_{n-1} > C\log p_n$? And what is the best known constant to date?

Edit: in view of the wikipedia entry on this topic, it seems that the correct thing to look at is

$$\displaystyle p_n - p_{n-1} > \frac{c \log n \log \log n \log \log \log \log n}{(\log \log \log n)^2}.$$ This seems like a rather unnatural function, but for now it is not known if the constant $c$ in the above inequality may be taken to be arbitrarily large. Is this inequality expected to be the right order of magnitude?

Thanks for any insight.