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

Step 2 of 8: The GPY sieve: a weighted count that forces primes to appear

In plain words

Imagine giving each integer $n$ in a big range a nonnegative 'weight' $w_n$ that is large exactly when $n+h_1,\dots,n+h_k$ have few small prime factors, so they are plausible candidates for being prime. If a cleverly weighted count of how many of the $n+h_i$ are prime, minus a target number $\rho$ , comes out positive overall, then some single $n$ in the range must actually contribute more than $\rho$ primes among $n+h_1,\dots,n+h_k$ .

Goldston, Pintz and Yıldırım built exactly such weights from the Möbius function $\mu$ , mimicking the classical Selberg sieve, and showed how to estimate the resulting sum precisely enough to find a positive contribution.

S(N,\rho)=\sum_{N\le n<2N}\Big(\sum_{i=1}^{k}\chi_{\mathbb{P}}(n+h_i)-\rho\Big)w_n,\qquad w_n=\Big(\sum_{\substack{d\mid \prod_{i=1}^k(n+h_i)\\ d<R}}\mu(d)\big(\log R/d\big)^k\Big)^2

Detailed analysis

For a fixed admissible $\mathcal{H}=\{h_1,\dots,h_k\}$ , nonnegative weights $w_n$ and a target $\rho>0$ , Maynard considers $S(N,\rho)=\sum_{N\le n<2N}\big(\sum_{i=1}^k \chi_{\mathbb{P}}(n+h_i)-\rho\big)w_n$ , where $\chi_{\mathbb{P}}$ is the indicator function of the primes (Maynard 2013, §2, eq. 2.1). If $S(N,\rho)>0$ for all large $N$ , some $n\in[N,2N)$ has weight $w_n>0$ and at least $\lfloor \rho+1\rfloor$ of $n+h_1,\dots,n+h_k$ prime — because $w_n\ge0$ everywhere, the only way the weighted sum can be positive is if the bracket is positive at some weighted point.

The classical GPY choice of weights is a Selberg-type square, $w_n=\big(\sum_{d\mid \prod_i(n+h_i),\,d<R}\mu(d)(\log R/d)^k\big)^2$ , where $\mu$ is the Möbius function and $R$ is a parameter controlling how far the divisor sum runs (Maynard 2013, §2, eq. 2.2). Squaring guarantees $w_n\ge0$ , and the specific choice of coefficients makes $S(N,\rho)$ computable via standard sieve technology, provided one knows how evenly primes are distributed in arithmetic progressions up to modulus around $R$ .

With this one-dimensional recipe, GPY could show gaps are infinitely often a vanishing fraction of $\log p_n$ , but not literally bounded — the weights treat the divisibility of the whole product $\prod_i(n+h_i)$ as one block, which turns out to be too rigid. The next step is Maynard's fix.

Terms in this step

Sieve weight: A nonnegative number $w_n$ attached to each integer $n$ in a range, designed so that $n$ contributes strongly to a counting sum exactly when $n+h_1,\dots,n+h_k$ have no small prime factors, mimicking primality.
Möbius function ( $\mu$ ): The arithmetic function with $\mu(1)=1$ , $\mu(n)=(-1)^r$ if $n$ is a product of $r$ distinct primes, and $\mu(n)=0$ if $n$ has a repeated prime factor; it is the classical tool for turning divisor sums into sieve estimates via inclusion–exclusion.