  Let $X$ be a large parameter. We will first give a new estimate for the
integral moments of primes in short intervals of the type $(p,p+h]$, where
$p\leq X$ is a prime number and $h=\odi{X}$. Then we will apply this to prove
that for every $\lambda>1/2$ there exists a positive proportion of primes
$p\leq X$ such that the interval $(p,p+ \lambda\log X]$ contains at least a
prime number. As a consequence we improve Cheer and Goldston's result on the
size of real numbers $\lambda>1$ with the property that there is a positive
proportion of integers $m\leq X$ such that the interval $(m,m+ \lambda\log X]$
contains no primes. We also prove other results concerning the moments of the
gaps between consecutive primes and about the positive proportion of integers
$m\leq X$ such that the interval $(m,m+ \lambda\log X]$ contains at least a
prime number. The last application of these techniques are two theorems (the
first one unconditional and the second one in which we assume the validity of
the Riemann Hypothesis and of a form of the Montgomery pair correlation
conjecture) on the positive proportion of primes $p\leq X$ such that the
interval $(p,p+ \lambda\log X]$ contains no primes.
