9 - Further Work and the Epilogue

Summary

Further results are flowing like a stream. By early 2018 the principal works which have formed the core of this book had collectively received over 210 citations registered on MathSciNet. In this chapter we summarize some of these results, depending also on the list of Andrew Granville. The material covered is but an indication of the fundamental nature of the breakthroughs which have been described, and much more is expected in the future. This chapter gives a description of the results but without proofs. The topics include bounded gaps for primes described by affine forms, clusters of primes in intervals, gaps between almost primes, clusters of primes in intervals, the set of limit points of the sequence of normalized consecutive prime differences, Artin’s primitive root conjecture, arithmetic progressions of primes with a fixed common difference, prime ideals inthe ring of integers of number fields, irreducible polynomials, the coefficients of a class of modular forms including Ramanujan’s tau-function form, quadratic twistsof a class of elliptic curves including the congruent number elliptic curve, and results obtained assuming the Elliott–Halberstam and generalized Elliott–Halberstam conjectures. The results often take the form of establishing bounded gaps for primes of a particular form.