Worked solution: Maynard's multidimensional sieve: bounded gaps between primes (2013)

Step 3 of 8: Maynard's fix: let the weight see each shift separately

In plain words

The GPY weight only cared about divisors $d$ of the whole product $(n+h_1)\cdots(n+h_k)$ , treating the $k$ shifted numbers as one indivisible block. Maynard's key new idea is to instead let the coefficient depend on a whole tuple of divisors $(d_1,\dots,d_k)$ , one $d_i$ for each shifted number $n+h_i$ separately.

This sounds like a small change, but it is like upgrading from a one-dimensional dial to a $k$ -dimensional control panel: it gives vastly more freedom to design weights that favor $n$ 's where many of the $n+h_i$ look prime simultaneously, rather than just where the product as a whole looks favorable.

w_n=\Big(\sum_{d_i\mid n+h_i\ \forall i}\lambda_{d_1,\dots,d_k}\Big)^2

Detailed analysis

Maynard's new weight is $w_n=\big(\sum_{d_i\mid n+h_i\,\forall i}\lambda_{d_1,\dots,d_k}\big)^2$ , summed over all $k$ -tuples of divisors $(d_1,\dots,d_k)$ with $d_i\mid n+h_i$ for every $i$ (Maynard 2013, §2, eq. 2.4). Unlike GPY's $\lambda_d$ , which was a single-variable coefficient, the new coefficient $\lambda_{d_1,\dots,d_k}$ is a genuinely $k$ -dimensional array — hence 'multidimensional sieve'.

Maynard notes this idea was not entirely without precedent (Selberg had suggested something similar for twin primes, and Goldston–Yıldırım had tried a restricted version), but earlier attempts capped each $d_i$ at $R^{1/k}$ , a restriction so severe it erased the benefit. Maynard's choice instead lets the coefficients roughly factor as $\lambda_{d_1,\dots,d_k}\approx\big(\prod_i \mu(d_i)\big) f(d_1,\dots,d_k)$ for a smooth function $f$ of $k$ real variables, with $d_i$ individually allowed to range up to $R$ (Maynard 2013, §2, eq. 2.5).

This multidimensional freedom is precisely what removes the earlier barrier: the counting sums built from these weights can now be evaluated (Proposition 4.1 below) in terms of an arbitrary smooth function $F$ of $k$ variables, turning sieve design into a calculus-of-variations problem the next steps will exploit.

Terms in this step

Multidimensional sieve weight: A sieve coefficient $\lambda_{d_1,\dots,d_k}$ indexed by a $k$ -tuple of divisors, one per shift $n+h_i$ , rather than by a single divisor of the whole product — the extra degrees of freedom are what make Maynard's method more powerful than GPY's.