TLDR: I have two questions thanks in advance:
- Can we assume that the $w_{n}$ are smooth approximations and if so why? I've always struggled in knowing when it is permissible / how to find smooth approximations
- Is it correct that in the Maynard case the probability distribution that is found is $\mathbb{P}(n+h_{i}\text{ is prime}) \asymp \frac{\log k}{k}$? I came to this by looking at equation 8.19 of the Maynard paper (last page) and corollary 6.4 (page 44) of the following paper: https://arxiv.org/pdf/1407.4897.pdf
Further detail:
Having looked at this more I've found the following:
The overall goal is we want to find weights $w_{n}$ to be larger than but as close as we can get to $\mathbb{1}_{\text{all $n + h_{i}$ prime}}$
It is best to have deviations between our chosen weights $w_{n}$ and $\mathbb{1}_{\text{all $n + h_{i}$ prime}}$ when we have many of the $n+h_{i}$ being prime because then we will have $\mathbb{P}(n + h_{i} \text{is prime})$ being large for each $i$.
In the GPY case we are considering Selberg type weights of the form $w_{n} = \Big(\sum_{d|n}\lambda_{d}\Big)^{2}$ where $\lambda_{d} = \mathbb{1}_{d \leq R}\mu(d)F(\frac{\log R}{d})$ where $F$ is a polynomial. In other words, the GPY weights are like the square of $$\sum_{d|\prod(n+h_{i})}\mu(d)\log\Big(\dfrac{\prod_{i=1}^{k}(n+h_{i})}{d}\Big)^{k}$$
In the Maynard case we are considering weights that are like the square of $$\Big(\sum_{d_{1}|n+h_{1}}\mu(d_{1})\dfrac{\log(n+h_{1})}{d_{1}}\Big)\dots\Big(\sum_{d_{k}|n+h_{k}}\mu(d_{k})\dfrac{\log(n+h_{k})}{d_{k}}\Big)$$ for $k$ sufficiently large.
The above GPY weights vanish when not all of the translates, $n+h_{i}$ are prime.
The above Maynard weights vanish when not all of the translates, $n+h_{i}$ are prime powers.
So if we view the $w_{n}$ as a smooth approximation to the square of the expression above in the Maynard case. Then if the first factor was not a prime power (but the remaining factors were) it would vanish. If we were to approximate it using $w_{n}$ it would be a small factor, but would be compensated for by larger values at the remaining translates.
We also have that $w_{n}$ is larger when more and more of the $n+h_{i}$ are prime.
However in the GPY case even if we again view the $w_{n}$ as a smooth approximation we don't get this compensation of by larger values for the remaining translates... Furthermore since GPY found $\mathbb{P}(n+h_{i} \text{is prime}) \asymp \frac{1}{k}$ the smooth weights $W_{n}$ will be getting worse as $k$ gets larger.
Whereas in the Maynard case it looks like we can save a factor of $\log k$ and we have $\mathbb{P}(n+h_{i}\text{ is prime}) \asymp \frac{\log k}{k}$ and thus this tends to infinity as $k$ gets larger which is what we want.