Modern Sieve Applications

Modern sieve theory combines classical upper/lower bound sieves with additional input from algebraic geometry, harmonic analysis, and L-functions to achieve results beyond the parity barrier. The breakthroughs on gaps between primes exemplify this synthesis.

GPY Sieve and Bounded Gaps

Definition7.5GPY Method

The Goldston-Pintz-Yildirim (GPY) method uses a modified Selberg sieve with weights concentrated on integers $n$ such that several translates $n + h_1, \ldots, n + h_k$ are simultaneously prime. The key quantity is $S_k = \sum_{x \leq n \leq 2x} w(n) \left(\sum_{i=1}^k \mathbf{1}_{n+h_i \text{ prime}} - m\right)$ where $w(n)$ is a smooth Selberg-type weight and $m$ is a threshold. If $S_k > 0$ , then some tuple $(n+h_1, \ldots, n+h_k)$ contains at least $m + 1$ primes.

ExampleMaynard-Tao Theorem on Bounded Gaps

Theorem (Maynard, 2013; Tao, polymath8b): $\liminf_{n\to\infty}(p_{n+1} - p_n) \leq 246$ . More generally, $\liminf(p_{n+m} - p_n) < \infty$ for every fixed $m$ , i.e., there are bounded gaps containing $m+1$ primes. The method uses multidimensional Selberg sieve weights $w(n) = (\sum_{\substack{d_1 | n+h_1, \ldots, d_k|n+h_k \\ d_i \leq R}} \lambda_{d_1,\ldots,d_k})^2$ optimized over a smooth function of $k$ variables.

Sieve and Algebraic Input

Definition7.6Friedlander-Iwaniec and Algebraic Sieves

Friedlander-Iwaniec (1998): There are infinitely many primes of the form $a^2 + b^4$ . The proof combines the half-dimensional sieve (sieving a thin sequence with density $x^{3/4}$ ) with bilinear form estimates using algebraic geometry over $\mathbb{Z}[i]$ . This broke the parity barrier for a specific thin sequence by exploiting the algebraic structure of Gaussian integers.

RemarkGreen-Tao and Primes in Arithmetic Progressions

The Green-Tao theorem (2004): the primes contain arbitrarily long arithmetic progressions. The proof uses Szemerédi's theorem relativized to a pseudorandom set, with the pseudorandomness verified via sieve theory (the $W$ -trick and Goldston-Yildirim type estimates). This shows that sieve methods, combined with additive combinatorics, can prove results far beyond the classical sieve framework.

Numerical Sieve Applications

ExampleChen's Theorem

Chen's theorem (1966/1973): Every sufficiently large even number $N$ is the sum of a prime and a number that is either prime or a product of two primes ( $P_2$ ). The proof uses the weighted sieve (Rosser-Iwaniec linear sieve) with Bombieri-Vinogradov as input: $|\{p \leq N : N - p \in P_2\}| \gg \frac{N}{(\log N)^2}$ . The sieve lower bound barely beats zero thanks to the level of distribution $x^{1/2-\varepsilon}$ from BV.