Short intervals containing a prescribed number of primes

TL;DR

This paper proves that for any fixed number of primes, there are many short intervals of the form [n, n + λ log n] containing exactly that many primes, with results improving previous work for small λ.

Contribution

It establishes the existence of many short intervals with a prescribed number of primes for small λ, extending prior results in prime distribution.

Findings

Many short intervals contain exactly m primes for fixed m.

The number of such intervals is proportional to x / log x.

Results hold for sufficiently large x and small λ.

Abstract

We prove that for every nonnegative integer $m$ there exists an $\varepsilon>0$ such that if $\lambda\in (0,\varepsilon]$ and $x$ is sufficiently large in terms of $m$, then the number of positive integers $n\leq x$ 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.