6 - Maynard’s Radical Simplification

Summary

It is fascinating to see that while Polymath8a was making improvements to Zhang’s method, James Maynard and Terence Tao, independently using combinatorial and analytic methods respectively, were using a completely different approach to study bounded gaps. This was based on an idea suggested by Selberg, and is called the multidimensional sieve. Tao was later to incorporate his method into the work of Polymath8b reported in Chapter 8, while Maynard’s work is given in this chapter. Three sections are devoted to developing properties of the sieve. Then a simplified form of the derivation of an essential integral formula is given. After detailing Maynard’s optimization procedure, and his Rayleigh quotient-based algorithm and efficiency enhancing integral formulas, we give the proofs that the bound for an infinite number of prime gaps is not more than 600, that there are bounded gaps for an arbitrary preassigned finite number of primes.We show that the constants in Maynard’s choice of upper bound for several results are optimal among bounds of the given form. Finally, using great combinatorial counting, we give Maynard’s proofthat prime k-tuples have positive relative density if counted appropriately.