One significant result of the project has been the proof of the famous Duffin-Schaeffer conjecture from Diophantine approximation by the PI and his collaborator Dimitris Koukoulopoulos from Universite de Montreal. This problem was known as one of the most important questions in the area of metric Diophantine approximation, but had resisted proof for 75 years. In the original proposal for PRIMES some initial work by the PI and collaborators had indicated that a more combinatorial perspective from analytic number theory might be able to offer new insights into this problem. Fortunately this succeeded in roughly the way envisaged in the proposal – we solved a ‘model problem’ which had identified a key obstruction to previous approaches by developing a new technique inspired by graph theory. Having solved the model problem, the new technique could then be generalised to solve the original conjecture in full. This completely solved the very ambitious Objective 7, and opened up new methods for attacking many other related problems, which are currently being investigated by the PI, collaborators and other researchers in the field. This result has received a significant amount of positive feedback from the field.
Another significant result of the project has been to establish new 'zero density estimates' for the Riemann Zeta function, in joint work with Lary Guth from MIT. Zero density estimates can serve as substitutes for the Riemann Hypothesis for several questions, but many results on the distribution of primes have been limited by our inability to rule out the possibility of there being 'many' zeros with real part close to 3/4. This is a problem that we were stuck on for about 80 years, but this new work overcomes this obstacle (with corresponding new results for primes). An important ingredient in the proof was developing methods which were sensitive to the vertical distribution of zeros of the Riemann Zeta function, which was the key idea behind Objective 3.
A third major result of the project has been work by the PI and Kevin Ford from the University of Illinois at Urbana-Champaign on prime detecting sieves. We have introduced a new 'sieve method' for detecting primes from certain arithmetic information. This outperforms various previous competing sieves, but also has the feature of being provably optimal in various regimes. In particular, this has allowed us to explicitly construct sets satisfying arithmetic 'Type I/II' bounds in certain ranges but containing no primes. This puts a hard limit on current techniques, but also opens up the possibility of incorporating new ideas to distinguish such sets from sets of interest.
A fourth result has been a number of improvements on our understanding of primes in arithmetic progressions, which was an unanticipated set of results which came about from when the PI was working on a different topic. In a sequence of lengthy papers the PI produced new results which went beyond pioneering work of Bombieri-Friedlander-Iwaniec from the 80s, and proved stronger results about primes in arithmetic progressions. This has led to a spark of activity in the field, with several new arithmetic results on primes and prime-like objects following from this, as well as more refined results linking the spectral theory of automorphic forms to the distribution of primes.
A fifth result has been work of two of the postdoctoral team members employed on the project, Lasse Girmmelt and Jori Merikoski. They have developed a 'Poincare series approach' to various questions on arithmetic sums which are connected to certain 2-by-2 matrix groups. This provides an alternative means of obtaining several of the most important results on using the spectral theory of automorphic forms in arithmetic applications. Moreover, in several situations this approach can be made simpler and more transparent, with fewer technical calculations along the way. This additional clarity has allowed for obtaining improvements to various arithmetic questions, but this work also offers up the possibility of this potentially being an approach in more general situations where the picture is much more complicated.
