It is a long-standing unsolved problem to prove that there is always a prime between and . This is equivalent to showing that . Since the average size of is , and it is conjectured that , 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 .
Problem: Find a believable conjecture about the zeros of the -function which implies that for all . Even the case 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 holds in some neighborhood of then . The proof only requires the continuity of at . 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 such that for a positive proportion of primes . (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 such that for a positive proportion of .
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