The infamous twin prime conjecture states that there are infinitely many pairs of distinct primes which differ by 2. In April 2013, Yitang Zhang proved the existence of a finite bound \( B \) such that there are infinitely many pairs of distinct primes which differ by no more than \( B \). Zhang even showed that one can take B = 70000000. In November 2013, inspired by Zhang's extraordinary breakthrough, James Maynard dramatically slashed this bound to 600, by a substantially easier method. Both Maynard and Terry Tao, who had independently developed the same idea, were able to extend their proofs to show that for any given integer \( m \geq 1\) there exists a bound \( B_m\) such that there are infinitely many intervals of length \( B_m\) containing at least \( m\) distinct primes. We prove this herein, even showing that one can take \( B_m=e^{8m+5}\). If Zhang's method is combined with the Maynard-Tao setup, then it appears that the bound can be further reduced to 246. This article introduces these results, and explain the arguments that allow them to prove their spectacular results. The second half develops a proof of Zhang's main novel contribution, an estimate for primes in relatively short arithmetic progressions.

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We are interested in classifying those sets of primes \( \mathcal P\) such that when we sieve out the integers up to \( x\) by the primes in \( \mathcal P^c\) we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length \( x\) with primes including some in \( (\sqrt{x},x] \) , using methods motivated by additive combinatorics.

We give bounds on the number of distinct differences \( N_a-a\) as \( a\) varies over all elements of a given finite set \( A \subset (\mathbb R/\mathbb Z)^d,\ d\geq 1\) and \( Na\) is a nearest neighbour to \( a\) .

As long as people have studied mathematics, they have wanted to know how many primes there are. Getting precise answers is a notoriously difficult problem, and the first suitable technique, due to Riemann, inspired an enormous amount of great mathematics, the techniques and insights permeating many different fields. In this article we will review some of the best techniques for counting primes, centering our discussion around Riemann's seminal paper. We will go on to discuss its limitations, and then recent efforts to replace Riemann's theory with one that is significantly simpler.

In 2009 we wrote a screenplay Mathematical Sciences Investigation (MSI) ``The anatomy of integers and permutations'' based on analogies between the mathematical structure of integers and of permutations. This was then performed as a ``staged reading'', with professional actors, at the Institute for Advanced Study, at MSRI and at the 2012 CMS annual meeting. The screenplay attempts to describe some mathematical research, as well as to illustrate the process of researching in mathematics, in the context of a narrative told in a screenplay. The screenplay encompasses details usually considered far too technical for expository writing, by explaining them through metaphor. Our objective is to reach a wider audience than is usual for mathematical exposition. We invited a noted experimental stage designer, Michael Spencer, to visually enhance these metaphors by helping us to stage the reading of the screenplay in an evocative manner. We also collaborated with the musician, Robert Schneider, to write a mathematically based score for one particular scene. The screenplay is currently in development as a graphic novel, being drawn by the artist Robert Lewis. In this article we attempt to outline the challenges and issues, creatively and mathematically, that emerged through the unusual collaboration during the development of this project.

We determine for what proportion of integers \( h\) one now knows that there are infinitely many prime pairs \( p, p+h\) as a consequence of the Zhang-Maynard-Tao theorem. We consider the natural generalization of this to \( k\)-tuples of integers, and we determine the limit of what can be deduced assuming only the Zhang-Maynard-Tao theorem.