Recent results on large gaps between primes

Abstract

One of the themes of this paper is recent results on large gaps between primes. The first of these results has been achieved in the paper [12] by Ford, Green, Konyagin and Tao. It was later improved in the joint paper [13] of these four authors with Maynard. One of the main ingredients of these results are old methods due to Erd\H{o}s and Rankin. Other ingredients are important breakthrough results due to Goldston, Pintz and Yildirim [15, 16, 17], and their extension by Maynard on small gaps between primes. All these previous results are discussed shortly. The results on the appearance of $k$-th powers of primes contained in those large gaps, obtained by the author in joint work with Maier [23, 24, 25] are based on a combination of the results just described with the matrix method of Maier.