Elchin Hasanalizade: Some applications of the Goldston-Pintz-Yi ldi ri m sieve

The twin prime conjecture — that there exist infinitely many pairs of “twin primes” p, p+2 — is among the most famous open problems in number theory.

Recently, Goldston, Pintz and Y ildi r im (GPY) made a major and unexpected breakthrough in this direction using a simple variant of the Selberg sieve. We will discuss some of the key concepts behind the GPY method and results on primes in tuples and prime gaps in general.

In particular, we will discuss the notion of level of distribution, 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 will present a new result related to a conjecture of Erdos-Mirsky concerning the divisor function at consecutive integers.