4 - The Breakthrough of Goldston, Motohashi, Pintz and Yildirim

Summary

Goldson, Motohashi, Pintz and Yildirim’s work was the first major breakthrough for prime gaps. They showed that for each given proportion of the average gap, no matter how small, there were an infinite number of prime pairs distant apart less than that proportion. This was a major result, highly creative and technical, and rightly celebrated in the mathematical world and more broadly. This chapter provides a complete exposition of their work, giving background on how their ideas evolved, the basic structures (admissible tuples, truncated von Mongoldt functions, using two tuples simultaneously, extending tuples, using Gallagher’s asymptotic result, and using the Elliott–Halberstam conjecture to show the potential best gap size (i.e., 16) between consecutive primes, and that any improvement on Bombieri–Vinogradov would result in bounded gaps between primes. Each of the preliminary and fundamental lemmas are proved, as is the full proof and a simplified proof. To show how it all fits together, there is an overview and flow diagram. The methods are elementary, but intricate, the most being possibly an estimate based on a double complex contour integral. It is fair to say the work inspired, at least in part, each of the breakthroughs which followed, although only Zhang and Maynard used one or two of their results explicitly.