By a 1985 result of Maier, the asymptotic $$\pi(x+h(x))-\pi(x) \sim \frac{h(x)}{\log x} \ (x \to \infty)$$ fails if $h(x)$ is any power of $\log x$. Are there known examples of functions $h(x) = o(x)$ with $(\log x)^a = O(h(x))$ for all $a>0$ for which the asymptotic above fails? If so, what is the best such result known, in the sense that the order of growth of $h(x)$ is largest? For example, is it known to fail for $h(x) = e^{(\log \log x)^t}$ for some $t >1$?
Reference: H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985) 221-225.
There are certainly not known examples, as far as I know. I think it's reasonable to conjecture that there will be no larger examples. The conjectures for the ratio $$ \frac{\pi(x+(\log x)^A)-\pi(x)}{(\log x)^A/\log x} $$ is that its lim sup $M(A)$ and its lim inf $m(A)$ both approach $1$ as $A\to\infty$, which to me suggests that the corresponding limit will equal $1$ as soon as "$A$" goes to infinity with $x$, that is, as soon as $h(x)$ is not $(\log x)^{O(1)}$.