I am going through James Maynard's paper, *Small Gaps between Primes*, and have a number of questions regarding his approach. First, I am wondering why uses weights in his approach. While I generally understand the meaning of (2.1) on pg. 3:
$$
S(N,\rho)=\sum\_{N\le n<2N}\Bigl(\sum\_{i=1}^k\chi\_{\mathbb{P}}(n+h\_i)-\rho\Bigr)w\_n, 
$$
where
$$
w\_n = \left(\sum\_{d\_i | n+h\_i \forall i} \lambda\_{d\_i,\ldots,d\_k}\right)^2 \mbox{ and } \lambda\_{d\_i,\ldots,d\_k} \approx \left( \prod\mu(d\_i)f(d\_1, \ldots, d\_k) \right),
$$
I am struggling to understand the role of these sieve weights. What is their purpose, and what do they do? How does use of such weights help Maynard's proof?


Second, I also wonder what is the purpose of $M\_k$, as defined in Proposition 4.2 on p.5. Why do we need it and what does it do? 


I read through the GPY paper, and also notice that they follow a similar approach to Maynard's, so looking there did not really help. 


PS. I apologise if my questions are inappropriate in any way for Mathoverflow. I am an undergraduate trying hard to understand this paper, and realised that this should be a good place to seek answers. 



Use of weights in the GPY's and Tao-Maynard's work on the twin prime conjecture
Very briefly, the role of the weights is to pick $n$'s for which the shifted admissible $k$-tuple $\{n+h\_1,\dots,n+h\_k\}$ has a better chance to contain at least two primes. Without any weighting the average number of primes lying in this tuple would be zero, so the weights are really there to 
counterbalance the fact that the primes have density zero.


For a more informative (or more technical) description of why the Maynard-Tao weights work so well, see my response to [this earlier MO post](https://mathoverflow.net/questions/229430/why-do-the-maynard-tao-weights-work-so-well). Hope this helps!


