Short intervals with a given number of primes

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

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