On smooth gaps between consecutive prime numbers

Abstract

Investigations concerning the gaps between consecutive prime numbers have long occupied an important position on the interface between additive and multiplicative number theory. Perhaps the most famous problem concerning these gaps, the Twin Prime Conjecture, asserts that the aforementioned gaps are infinitely often as small as 2. Although a proof of this conjecture seems presently far beyond our reach (but see [5] and [10] for related results), weak evidence in its favour comes from studying unusually short gaps between prime numbers. Thus, while it follows from the Prime Number Theorem that the average gap between consecutive primes of size about x is around log x , it is now known that such gaps can be infinitely often smaller than 0–249 log x (this is a celebrated result of Maier [12], building on earlier work of a number of authors; see in particular [7], [13], [3] and [11]). A conjecture weaker than the Twin Prime Conjecture asserts that there are infinitely many gaps between prime numbers which are powers of 2, but unfortunately this conjecture also seems well beyond our grasp. Extending this line of thought, Kent D. Boklan has posed the problem of establishing that the gaps between prime numbers infinitely often have only small prime divisors, and here the latter divisors should be small relative to the size of the small gaps established by Maier [12]. In this paper we show that the gaps between consecutive prime numbers infinitely often have only small prime divisors, thereby solving Boklan's problem. It transpires that the methods which we develop to treat Boklan's problem are capable also of detecting multiplicative properties of more general type in the differences between consecutive primes, and this theme we also explore herein.