A note on gaps

Hisanobu Shinya

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arXiv · 0809.3458

A note on gaps

Hisanobu Shinya

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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 $N_ε(x)$ the number of primes $\leq x$ which satisfy the inequality $d(p_{k}) \leq (\log p_{k})^{2 + ε}$, where $ε> 0$ is arbitrary and fixed, and by $π(x)$ the number of primes less than or equal to $x$. In this paper, we first prove a theorem that $\lim_{x \to \infty} N_ε(x)/π(x) = 1$. A corollary to the proof of the theorem concerning gaps between consecutive squarefree numbers is stated.

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BibTeX RIS

Hisanobu Shinya. 2011-09-11. A note on gaps. https://arxiv.org/abs/0809.3458

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