Bounded intervals containing a given number of primes

Abstract

Let $m$ be a non-negative integer and $f(x)$ a positive, non-decreasing function satisfying certain conditions. We give an explicit lower bound for the number of integers $n\leq x$ such that $\#([n, n+f(n)] \cap \mathbb{P})=m$ for sufficiently large $x$, where $\mathbb{P}$ denotes the set of prime numbers. This work extends the results of Mastrostefano and of Freiberg, and also makes the lower bound of Masrtrostefano explicit. In addition, we show that if the interval length is sufficiently large, then there exist infinitely many bounded intervals of the same length that contain exactly a prescribed number of primes.