A note on gaps

Abstract

Let $p_k$ denote the $k$-th prime and $d(p_k)=p_k-p_{k-1}$, the difference between consecutive primes. We denote by $N_{\epsilon }(x)$ the number of primes $\le x$ which satisfy the inequality $d(p_k)\le (\log p_k{)}^{2+\epsilon }$, where $\epsilon >0$ is arbitrary and fixed, and by $\pi (x)$ the number of primes less than or equal to $x$. In this paper, we first prove a theorem that ${\lim }_{x\to \infty }{N}_{\epsilon }(x)/\pi (x)=1$. A corollary to the proof of the theorem concerning gaps between consecutive squarefree numbers is stated.

arXiv commentThe paper should simply be forgotten, as it was pointed that this claim can be shown in three sentences