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I'm attempting to understand why the Maynard-Tao weights are successful in proving bounded gaps between primes, but the GPY weights are not.

These two posts do an excellent job in giving an overview of how one arrives at the Maynard-Tao weights and the role of the weights themselves:

The additional flexibility in allowing the $\lambda_{d_{1},\dots,d_{k}}$ to depend on the divisors $d$ individually allows the method to be succeed. If possible I'd like a deeper understanding why the additional flexibility allows it to succeed.

Soundararajan's expository paper: https://arxiv.org/pdf/math/0605696.pdf (pages 9 - 14) explain the core ideas of the GPY method and that the ratio of $$\dfrac{\Big(\sum_{\substack{x \leq n \leq 2x} \\ n + h_{j} \text{prime}}a(n)\Big)}{\Big(\sum_{x \leq n \leq 2x}a(n)\Big)}$$ cannot be made greater than $\frac{1}{k}$ and we therefore fail to just prove bounded gaps between primes.

Therefore the GPY method finds a probability distribution such that Probability$(n+h_{i} \text{is prime}) \asymp \frac{1}{k}$ and this differs from the optimal value by a factor of about $k.$

The Maynard-Tao weights clearly give a better weighting so this issue is overcome and find a different probability distribution to enable the inequality to hold but I'm a bit confused. Why is it that because the $\lambda_{d_{1},\dots,d_{k}}$ to depend on the divisors $d$ individually, this issue is overcome?

What is the probability distribution that the Maynard-Tao weights find such that the inequality holds?

Apologies if this isn't the appropriate place to ask this question. I'm an Undergraduate interested in analytic number theory...

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TLDR: I have two questions thanks in advance:

  1. 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
  2. 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.

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