8 - Variations on Bombieri–Vinogradov

Summary

The previous chapters represented a linear time based progression of ideas. This chapter departs from that sequence in several ways. Here we report on the enhancements of Zhang’s work made by Polymath8a which showed Zhang’s original prime gap bound of 70 million could be significantly reduced. They introduced new concepts and optimized over parameter and function spaces, but fell short of Maynard’s bound of 600. Their ideas are valuable – for example Polymath8a’s variations of Bombieri–Vinogradov based on multiple dense divisibility. This generalizes the smoothness requirement of Motohashi, Pintz and Zhang, imposing a weaker parametric condition, multiple dense divisibility, extracting the essence of smoothness.Polymath8a completed their work in an unbelievably short period of time. This chapter is not to be taken lightly and goes into great technical detail. There are major mathematical tools and intricate estimates. A complete proof of the derivation of their bound 14950 is given, including proofs of supporting material in appendices. This is not their best bound of 4680, which relies on results of Deligne and is outside the scope of this book. However, we do give information and references on how readers if necessary might access background material from algebraic geometry for example.