Bounds on gaps between primes

It is a long-standing unsolved problem to prove that there is always a prime between $n^2$ and $(n+1)^2$. This is equivalent to showing that $p'-p\le p^{1/2}$. Since the average size of $p'-p$ is $\log p$, and it is conjectured that $p'-p\ll \log^2 p$, current results seem to be very far from the final truth.

Goldston and Heath-Brown [ MR 85e:11064] have shown that the pair correlation conjecture implies $p'-p=o(p^{1/2}\log^{1/2}p)$.

Problem: Find a believable conjecture about the zeros of the $\zeta$-function which implies that $p'-p\ll p^A$ for all $A>0$. Even the case $A=\frac12$ would be significant.

For an example of a non-believable conjecture which may imply that there are small gaps between consecutive primes, see the article on the Alternative Hypothesis.

Heath-Brown [ MR 83m:10078] showed that if Montgomery's conjecture on $F(\alpha; T)$ holds in some neighborhood of $\alpha=1$ then $\liminf \frac{p'-p}{\log p} = 0$. The proof only requires the continuity of $F(\alpha; T)$ at $\alpha=1$. This continuity also follows from the alternative hypothesis, so there may be hope of proving this unconditionally.

Erdös used sieve methods to show that there exists $\delta>0$ such that $p'-p \le (1-\delta) \log p$ for a positive proportion of primes $p$. (It would be helpful if someone could provide details on the history of this problem and an up-to-date account of current results).

It has not yet been shown that there is a $\lambda>1$ such that $p'-p \ge \lambda \log p$ for a positive proportion of $p$.

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