关于差距的说明
A note on gaps
摘要：
令$p_{k}$表示第$k$个素数，$d(p_{k}) = p_{k} - p_{k - 1}$表示连续素数之间的差。我们用$N_ε(x)$表示满足不等式$d(p_{k}) \leq (\log p_{k})^{2 + ε}$的素数$\leq x$的个数，其中$ε> 0$是任意的和固定的，用$π(x)$表示小于或等于$x$的素数的个数。本文首先证明了一个定理，即$\lim_{x \to \infty} N_ε(x)/π(x) = 1$ .一个推论的定理证明之间的差距连续squarefree数的陈述。
英文摘要：
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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