Suppose $m$ is a positive integer. A quantity of interest is


$$
H\_m = \liminf\_{n\to\infty} \left(p\_{n+m} - p\_n \right)
$$


The twin prime conjecture, is, of course $H\_1 = 2$, the the prime k-tuples conjecture of Hardy and Littlewood asserts that $H\_2 = 6$, $H\_3 = 8$ and so on. Goldston-Pintz-Yildirim showed that under the Elliot-Halberstam conjecture, $H\_1 < \infty$, a result that Yitang Zhang has famously recently established unconditionally.


It is my understanding that the methods of GPY were essentially restricted when dealing with $m>1$. One of their non-trivial results was that $g\_n = p\_{n+1} - p\_n$ satisfies $g\_n = o(\log p\_n)$, or in other words that $\liminf \frac{g\_n}{\log p\_n} = 0$, which obviously trivially follows from Zhang's result.


However, it is my understanding that even under the full Elliot-Halberstam conjecture, GPY were unable to prove even their weaker result for $m\geq 2$. Maynard and Tao (see <http://arxiv.org/abs/1311.4600>) have shown that in fact, that $H\_m < \infty$. Apparently, this work only required the classical Bombieri-Vinogradov theorem, not even the variation of that Zhang's proof uses.


So my question is this: what shortcoming did the GPY approach have that did not allow it to extend to $m>2$, and how has Maynard's approach solved this problem?


PS: I apologize if this question is somehow inappropriate for Mathoverflow. I am an undergraduate trying to read up on these recent results by myself as much as I can, and I figured this would probably be the best place to ask my question.



What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$
The major difference is the choice of the Selberg Sieve weights. I strongly recommend reading [Maynard's paper](http://arxiv.org/abs/1311.4600). It is well written, and the core ideas are nicely explained. 




---


In what follows, I'll give a brief explanation of what Selberg Sieve weights are, why they appear, and what was chosen differently in Maynard's paper compared to that of Goldston Pintz and Yildirim. 


Let $\chi\_{\mathcal{P}}(n)$ denote the indicator function for the primes. Given an admissible $k$-tuple $\mathcal{H}=\{h\_1<h\_2<\cdots<h\_k\}$ the goal is to choose a non-negative function $a(n)$ so that 


$$\sum\_{x<n\leq 2x} \left(\sum\_{i=1}^k \chi\_{\mathcal{P}}(n)(n+h\_i)\right)a(n)\geq \rho \sum\_{x<n\leq 2x} a(n)\ \ \ \ \ \ \ \ \ \ (1)$$ 


for some constant $\rho>1$. If this is the case, then by the non-negativity of $a(n)$, we have that for some $n$ between $x$ and $2x$ $$\left(\sum\_{i=1}^k\chi\_{\mathcal{P}}(n)(n+h\_i)\right)\geq \lceil\rho \rceil,$$ and so there are at least $\lceil\rho \rceil$ primes in an interval of length at most $h\_k-h\_1$. This would imply Zhangs theorem on bounded gaps between primes, and if we can take $\rho$ arbitrarily large, it would yield the Maynard-Tao theorem. However, choosing a function $a(n)$ for which equation $(1)$ this holds is very difficult. Of course we would like to take $$a(n)=\begin{cases}
1 & \text{when each of }n+h\_{i}\text{ are prime}\\
0 & \text{otherwise}
\end{cases} ,$$
 as this maximizes the ratio of the two sides, but then we cannot evaluate $\sum\_{x<n\leq 2x} a(n)$ as that is equivalent to understanding the original problem. Goldston Pintz and Yildirim use what is known as *Selberg sieve weights*. They chose $a(n)$ so that it is $1$ when each of $n+h\_i$ are prime, and non-negative elsewhere, in the following way:


Let $\lambda\_1=1$ and $\lambda\_d=0$ when $d$ is large, say $d>R$ where $R<x$ will be chosen to depend on $x$. Then set $$a(n)=\left(\sum\_{d|(n+h\_1)\cdots(n+h\_k)} \lambda\_d\right)^2.$$ By choosing $a(n)$ to be the square of the sum, it is automatically non-negative. Next, note that if each of $n+h\_i$ is prime, then since they are all greater than $x$, which is greater than $R$, the sum over $\lambda\_d$ will consist only of $\lambda\_1$. This means that $a(n)=1$ when each of the $n+h\_i$ are prime, and hence $a(n)$ satisfies both of the desired properties. Of course, we do not want $a(n)$ to be large when none, or even very few of the $n+h\_i$ are prime, so we are left with the difficult optimization problem of choosing $\lambda\_d$. The optimization in the classical application of the Selberg sieve to bound from above the number of prime-tuples involves diagonalizing a quadratic form, leading to


$$\lambda\_d = \mu(d) \left(\frac{\log(R/d)}{\log R}\right)^k,$$ 


and so 
$$a(n)=\left(\sum\_{\begin{array}{c}
d|(n+h\_{1})\cdots(n+h\_{k})\\
d<R
\end{array}}\mu(d)\left(\frac{\log(R/d)}{\log R}\right)^{k}\right)^{2}.$$


This choice is not so surprising since the related function $\sum\_{d|n}\mu(d) \left(\log(n/d)\right)^k$ is supported on the set of integers with $k$ prime factors or less. Goldston Pintz and Yildirim significantly modified this, and used a higher dimensional sieve to obtain their results.


The critical difference in the approach of Maynard and Tao is choosing weights of the form


$$a(n)=\left(\sum\_{d\_{i}|(n+h\_{i})\ \forall i}\lambda\_{d\_{1}\cdots d\_{k}}\right)^{2}.$$ This gives extra flexibility since each $\lambda\_{d\_1,\dots,d\_k}$ is allowed depend on the divisors individually. Using this additional flexibility Maynard and Tao independently established equation $(1)$ for any $\rho>1$, with $k$ depending on $\rho$.




---


Here are four great resources where you can read more:


* Maynard's [paper](http://arxiv.org/abs/1311.4600).
* Tao's [blog posts](http://terrytao.wordpress.com/2013/11/19/polymath8b-bounded-intervals-with-many-primes-after-maynard/).
* Granville's [article](http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf) on the Maynard-Tao theorem and the work of Zhang.
* Soundararajan's [exposition](http://arxiv.org/abs/math/0605696) of Goldston Pintz and Yildirim's argument.


