Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville :


  “The big experts in the field had
  already tried to make this approach
  work,” Granville said. “He’s not a
  known expert, but he succeeded where
  all the experts had failed.”


What new approach did Yitang Zhang try & what did the experts miss in the first place?

My understanding of this, which is essentially cobbled together from the various news accounts, is as follows:

Let $\pi(x;q,a)$ denote the number of primes less than $x$ congruent to $a\bmod q$, and $\pi(x)$ the number of primes less than $x$. $\phi(n)$ is the number of positive integers less than or equal to n that are relatively prime to n, and is usually referred to as Euler's totient function. Denote by $EH(\theta)$ the assertion that $\forall A > 0, \exists C > 0, \forall x > 2$, the following inequality holds:

$$\sum_{1\leq q \leq x^{\theta}} \max_{\gcd(a,q)=1} \left| \pi(x;q,a) -  \frac{\pi(x)}{\phi(q)} \right| \ll \frac{x}{\log^A(x)} (*)$$
for all large $x$.

The Bombieri–Vinogradov theorem asserts that EH($\theta$) holds for $\theta <1/2$, and the Elliot–Halberstam conjecture asserts that EH($\theta$) holds for all $\theta<1$.

In the mid 2000's Goldston, Pintz and Yildirim proved that if the Elliott–Halberstam conjecture holds for any level of distribution $\theta>1/2$ then one has infinitely many bounded prime gaps (where the size of the gap is a function of $\theta$,  for $\theta>.971$ they get a gap of size 16). Since Bombieri–Vinogradov tells us that $EH(\theta)$ holds for $\theta < 1/2$, in some sense the Goldston–Pintz–Yildirim arguments just barely miss giving bounded gaps.

On the other hand, in the 1980s Fouvry and Iwaniec were able to push the level of distribution in the Bombieri–Vinogradov theorem above $1/2$ at the expense of (1) removing the absolute values, (2) removing the maximum over residue classes and, (3) weighting the summand by a `well-factorable' function. This was subsequently improved in a series of papers by Bombieri, Friedlander and Iwaniec.  For the Goldston–Pintz–Yildirim argument the first two restrictions didn't pose a significant hurdle, however the inclusion of the well-factorable weight appeared to prevent one from using it with the Goldston–Pintz–Yildirim machinery. 

This is where my knowledge becomes extremely spotty, but my understanding is that Zhang  replaces the Goldston–Pintz–Yildirim sieve with a slightly less efficient one. While less efficient, this modification allows him some additional flexibility. He then is able to divide up the quantity that is estimated using (*) into several parts. Some of these are handled using well-established techniques, however in the most interesting range of parameters he is able to reduce the problem to something that involves a well-factorable function (or something similar) which he ultimately handles with arguments similar to those of Bombieri, Friedlander and Iwaniec.

Zhang's argument reportedly gives a gap size that is around 70 million. My suspicion is that this gap will quickly decrease. In their theorem, Bombieri, Friedlander and Iwaniec give a level of distribution around $4/7$, where (according to these notes) Zhang seems to be working with a level of distribution of the form $1/2 + \delta$ for $\delta$ on the order of $1/1000$, so there is likely room for optimization. As a reference point if one had the level of distribution $55/100$ ($< 4/7$) in the unmodified Elliot–Halberstram conjecture (without the well-factorable weight), the Goldston, Pintz and Yildirim gives infinitely many gaps of size less than 2956.

The parity problem, however, is widely known to prevent approaches of this form (at least on their own) from getting a gap smaller than length 6. Of course, there are more technical obstructions as well and even on the full Elliott–Halberstam conjecture one can only get a gap of 16 at present.

 
