Primes in short intervals

Prime numbers are some of the most important and fundamental objects in mathematics, but unfortunately they remain poorly understood, even though they have been studied for thousands of years (going back at least as far as the ancient Greeks). Many of the most basic questions regarding their distribution are open problems which have resisted advances for hundreds of years. This project will investigate one of the key questions concerning the distribution of prime numbers: the gaps between them.

It is believed that the primes should have a very particular distribution in short intervals. There are numerous predictions (both from experimental and theoretical origins) about precisely how many primes there should be in intervals of different lengths, and this not only describes the distribution of primes on a medium scale, but also describes the distribution of gaps between primes. The objective of the project is to unconditionally prove new stronger results about the number of primes in different short intervals by developing and refining techniques for making progress on these famous conjectures.

There are two main sets of techniques for studying primes in short intervals: sieves and Dirichlet polynomials. Dirichlet polynomials are closely related to the Riemann zeta function and their behaviour is connected to questions similar to the Riemann Hypothesis. These techniques tend to be most effective at studying large-scale properties of the primes. Sieves, on the other hand, are combinatorial (rather than complex-analytic) objects which often struggle to detect primes on their own, but have the advantage of being exceptionally flexible, so they produce non-trivial partial results in many contexts where Dirichlet polynomial techniques seemed destined to fail.

This project aims to develop new refined tools to combine ideas coming from sieve methods with those associated to Dirichlet polynomials. This will enable the strengths of both approaches to be brought to bear on questions about primes in short intervals and gaps between primes, and allow one to make partial progress on some of the most famous central problems in prime number theory.

The potential impact of this project is very wide, since it deals with such basic ubiquitous objects. Prime numbers appear across not just mathematics, but also science and the real world. The correct functioning of many vital day-to-day cryptographic algorithms is based on the assumption that many of these fundamental conjectures about the prime numbers are true. Moreover, these questions are amongst some of the longest-standing open problems in pure mathematics. Therefore it is a completely central task for mathematicians to make progress on these conjectures, and any partial progress which is consistent with the conjectures themselves will have an impact not only on the analytic number theory community, but also much further afield as well.

This project falls within the EPSRC number theory research area.