On the ratio of consecutive gaps between primes

Abstract

In the present work we prove a common generalization of Maynard- Tao’s recent result about consecutive bounded gaps between primes and of the Erdős-Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60-year-old problem of Erdős, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively. This is proved in the paper in a stronger form that not only dn = pn+1 - pn can be arbitrarily large compared to dn+1 but this remains true if dnC1 is replaced by the maximum of the k differences dn+1,…, dn+k for arbitrary fix k. The ratio can reach c(k) times the size of the classical Erdős-Rankin function with a constant c(k) depending only on k.