5 - The Astounding Result of Yitang Zhang

Summary

How Yitang Zhang arrived at his breakthrough result showing bounded gaps between primes unconditionally is retold in part in this chapter. GPY (Chapter 4) showed that any advance of the parameter in EH beyond one half was sufficient. In making his wonderful breakthrough, Zhang showedthat the full strength of EH beyond one half was not needed. Instead of summing over all moduli one could restrict their values to being smooth integers. In a tour de force of advanced methods, he showed that this restricted extension was unconditionally true for an explicit value of the parameter. No one before Zhang believed it to be possible to carry this through. Zhang used the methods of others, but significantly extended them to obtain sufficient flexibility to attain bounded gaps. The detail of the final part of Zhang’s argument is set out in this chapter, showing how he derived the bound. He chose explicit constants ending up with a gap size of 70 million. The ideas and methods used by Zhang come from Bombieri, Deligne, Deshouillers, Fouvry, Friedlander, Goldston, Heath-Brown, Iwaniec, Kuznetsiv, Motohashi, Pintz, Vinogradov, Yildirim and Weil and there are references given for much of this work.