János Pintz abstract

Small gaps between primes, primes in arithmetic progressions and arithmetic progressions of primes.

Our knowledge about small gaps between consecutive primes increased dramatically in the last decade. The famous twin prime conjecture asserts that the smallest distance is infinitely often equal to two. However, ten years ago we only knew that 1/4 of the average gap size of logp appeared infinitely often between two consecutive primes p and p'. In 2005 together with D. Goldston and C. Yildirim we proved in 2005 that 1/4 can be substituted by any positive c. Soon after this we showed that distances essentially less than the square root of the average distance occur infinitely often. Finally, during the last 12 months three mathematicians developed our method further: first Y. Zhang, and a few months later J. Maynard and T. Tao independently (using another generalization of our method) independently showed the existence of infinitely many bounded gaps between primes. The Polymath8B project of T. Tao refined Maynard's method in order to reach distances of at most 246 infinitely often between consecutive primes.In the lecture I will give an overview of the results and of the methods used to reach these results. I will also mention several further consequences of these methods in relation to the distribution of Polignac numbers (even integers which can be written in infinitely many ways as the difference of two consecutive primes), to the Hardy-Littlewood prime tuples conjecture, and to some conjectures of Erdős about gaps between consecutive primes. Finally I will mention a common generalization of the Green-Tao theorem and the above mentioned theorem of Zhang-Maynard-Tao, according to which there are arbitrarily long (finite) arithmetic progressions in the sequence of generalized twin primes (prime pairs of type p and p+d, with a fixed even d).