7 - Polymath’s Refinements of Maynard’s Results

Summary

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.