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.

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    $\begingroup$ I think Maynard's paper is written in such a way that minimal background is needed to really understand it (it's also 'elementary' in the sense that the proof works assuming any positive level of distribution for primes, i.e., any appropriately strong version of Bombieri-Vinogradov), but maybe the notation is too much for a first read through. It might help to first read some books on sieve theory, for example George Greave's book or Opera de Cribro (the latter does talk about the problem of bounded gaps between primes, up to the GPY development. The book predates Yitang Zhang's breakthrough) $\endgroup$ – Stanley Yao Xiao May 26 '18 at 15:34
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    $\begingroup$ I also recommend finding the related articles on Terry Tao's blog; they should add a lot of perspective. $\endgroup$ – Greg Martin May 26 '18 at 18:10
  • $\begingroup$ Does phrasing it this way help? He (and GPY, ...) is applying the probabilistic method to produce primes, and, as is usual in the probabilistic method, one has to choose a very smart probability distribution with respect to which to take the expectation to get it to be large (but also calculable). His weights precisely give such a distribution. $\endgroup$ – alpoge May 27 '18 at 0:12

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. Hope this helps!

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  • $\begingroup$ Yes, now I understand. Thank you very much! $\endgroup$ – Sultan Aitzhan May 28 '18 at 6:29

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