  A well-known conjecture asserts that, for any given positive real number
$\lambda$ and nonnegative integer $m$, the proportion of positive integers $n
\le x$ for which the interval $(n,n + \lambda\log n]$ contains exactly $m$
primes is asymptotically equal to $\lambda^me^{-\lambda}/m!$ as $x$ tends to
infinity. We show that the number of such $n$ is at least $x^{1 - o(1)}$.
