A note on gaps

2008/09/19 by Shinya, Hisanobu
#11B05 #FOS: Mathematics #General Mathematics (math.GM)

paper · doi:10.48550/arxiv.0809.3458

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 ≤ x which satisfy the inequality d(p_k) ≤ (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 → ∞} 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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