The Goldston-Pintz-Yildirim sieveand some applications

Abstract: The twin prime conjecture - that there exist infinitely many pairs of "twin primes" p, p + 2 - is among the most famous problems in number theory. In 2005, Goldston, Pintz and Yildirim (GPY) made a major and unexpected breakthrough in this direction using a simple variant of the Selberg sieve. Namely, they proved that the gap between consecutive primes is infinitely often arbitrarily smaller than average. In this master’s thesis we discuss some of the key concepts and ideas behind the GPY method and results on primes in tuples and prime gaps in general. In particular, we discuss the notion of the level of distribution of the primes, and in connection with this the celebrated Bombieri-Vinogradov theorem, which gives a result of the strength of the Generalized Riemann Hypothesis in an average sense.   As an application of the GPY method, we present a new result related to conjecture of Erdos-Mirsky concerning the divisor function at consecutive integers. This result is an improvement of Hildebrand’s previous work. .