Cramér's random model leads us to expect that the
primes are distributed in a Poisson distribution around their
mean spacing. It is conjectured that, for any given positive
number λ and nonnegative integer m, the proportion of
n ≤ x for which (n,n + λ logn] contains exactly m
primes is asymptotically equal to λme−λ/m! as
x → ∞. It is also conjectured that, for any given numbers
b > a ≥ 0, the proportion of n ≤ x for which
dn/logn ∈ (a,b], where dn = pn+1 − pn and pn is
the nth smallest prime, is asymptotically equal to
∫ab e−t dt as x → ∞.
By combining an Erdős–Rankin type construction, which
produces large gaps between consecutive primes, with the
Maynard–Tao breakthrough on bounded gaps between primes, we are
able to show that the number of n ≤ x for which
π(n + λlogn) − π(n) = m is at least x1 − o(1).
We are also able to show that at least 25% of nonnegative
real numbers are limit points of the sequence (dn/logn) of
normalized level spacings in the primes.
This includes joint work with William Banks and James Maynard.
(Translated from LATEX by HEVEA)