On the Ratio of Consecutive Gaps Between Primes

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 d n = p n + 1 − p n $$d_{n} = p_{n+1} - p_{n}$$ can be arbitrarily large compared to d n+1 but this remains true if d n+1 is replaced by the maximum of the k differences d n + 1 , … , d n + k $$d_{n+1},\ldots,d_{n+k}$$ 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.