A note on gaps

TL;DR

This paper proves that almost all primes have gaps smaller than a certain logarithmic bound, showing that the proportion of such primes approaches 100% as numbers grow large.

Contribution

It establishes that the ratio of primes with small gaps to all primes tends to one, and connects this result to properties of squarefree numbers.

Findings

Almost all primes satisfy the gap inequality as x approaches infinity.

The ratio N_ε(x)/π(x) converges to 1 for any fixed ε > 0.

A related corollary involves gaps between squarefree numbers.

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 $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 + \epsilon}$, 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.