Short intervals containing a prescribed number of primes

We prove that for every nonnegative integer $m$ there exists an $\epsilon >0$ such that if $\lambda \in (0,\epsilon ]$ and $x$ is sufficiently large in terms of $m$, then the number of positive integers $n\leq x$forwhichtheinterval$[n,n+\lambda\log n]$containsexactly$m$ primes is at least a constant times $x\mathrm{/}\log x\mathrm{.}$ This improves a result of T. Freiberg, when $\lambda$ is small.