Short intervals containing a prescribed number of primes

Abstract

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 $n\le x$ for which the interval $[n,n+\lambda \log n]$ contains exactly $m$ primes is at least a constant times $x/\log x.$ This improves a result of T. Freiberg, when $\lambda$ is small.

arXiv comment8 pages