Short intervals with a given number of primes

Abstract

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 ${\lambda }^m{e}^{-\lambda }/m!$ as $x$ tends to infinity. We show that the number of such $n$ is at least $x^{1-o(1)}$.