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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