The closest thing we have to a general method for finding primes in sets is to use sieve methods to turn the problem into some other (hopefully easier) arithmetic questions about the set.

Unfortunately this process is still poorly understood - we don’t know ‘how much’ arithmetic information is sufficient to guarantee the existence of primes, and how much is not sufficient. Often arguments are rather ad-hoc.

I’ll talk about work-in-progress with Kevin Ford which shows that many of our common techniques are not optimal and can be refined, and in many cases these new refinements are provably optimal.

Hoheisel used zero-density results to prove that for all x large enough there is a prime number in the interval $[x−x^{\theta}, x]$ with $θ < 1$. The connection between zero-density estimates and primes in short intervals was explicitly described in the work of Ingham in 1937. The approach of Ingham combined with the zero-density estimates of Huxley (1972) provides us with the distribution of primes in $[x−x^{\theta}, x]$ with $\theta > 7/12$. Further improvement upon the value of \theta was achieved by combining sieves with the weighted zero-density estimates in the works of Iwaniec and Jutila, Heath-Brown and Iwaniec, and Baker and Harman. The last work provides the best result achieved using zero-density estimates. We will discuss the main ideas of the paper by Baker and Harman and simplify some parts of it to show a more explicit connection between zero-density results and the sieved sums, which are used in the paper. This connection will provide a better understanding on which parts should be optimised for further improvements and on what the limits of the methods are. This project is still in progress.