This chapter reports in detail on some of the main contributions of Polymath8b, with a summary of their other results in an end note. They both completed and improved on Maynard using completely independent methods, and obtained wide-ranging results. For example, deriving bounds replacing asymptotic formulas for principal sums and then using that flexibility to complete a theorem proof, they showed that optimizations could be made without loss over symmetric functions, and derived a simple analytic upper bound revealing a limit to Maynard’s method. This chapter also reports in detail how they perturbed the standard simplex in a simple manner to derive the prime gap best current bound of 246. We give an improvement of the bound on this method which tends to the earlier bound as the parameter goes to zero. Overall, their methods based on Fourier analysis are simpler than those of Maynard. For example there is their alternative proof of “Maynard’s lemma” which gives a sufficient condition for a given number of primes in an infinite number of shifted admissible tuples of given size. There is also a discussion of Polymath8b’s algorithm and Bogaert’s Krylov basis method, both of which are included in PGpack.

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.