Besfort Shala

Application of sieve methods to gaps between primes

Sieve methods are a flexible set of tools in number theory, the aim of which is to describe the size of "sifted" sets of integers. The basic idea originates from the ancient Greek mathematician Eratosthenes, who invented a "sifting" algorithm to find prime numbers. This idea was substantially developed further only in the 20th century. The aim of this expository talk is to give a rather quick probabilistically flavoured introduction to sieve methods and Selberg's sieve in particular, and then focus on an application to gaps between primes. The latter is a result of the intimate connection between gaps between primes and counting k-tuples of prime numbers. We will give an overview of the key ideas of the method of Goldston, Pintz and Yıldırım (GPY), improved upon by Maynard, and briefly remark on the current state of the art on gaps between primes.