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| Maynard sieve | |
|---|---|
| Name | Maynard sieve |
| Field | Number theory |
| Introduced | 2013 |
| Contributors | James Maynard |
| Related | Prime number theorem, Selberg sieve, Goldston–Pintz–Yıldırım method, Elliott–Halberstam conjecture, Polignac's conjecture |
Maynard sieve is a sieve method in analytic number theory introduced by James Maynard in 2013 that produced striking advances on gaps between prime numbers and clustering of primes. The method built on ideas from the Goldston–Pintz–Yıldırım method and the Selberg sieve to obtain bounded gaps and dense prime constellations, influencing work on the Twin Prime Conjecture, Polignac's conjecture, and refinements of the Prime k-tuples conjecture. It rapidly inspired further results by researchers at institutions including University of Oxford, University of Toronto, Princeton University, Institute for Advanced Study, and Massachusetts Institute of Technology.
Maynard's approach adapted combinatorial sieve weights and optimization techniques from Selberg sieve constructions and connected them to advanced distributional estimates like the Bombieri–Vinogradov theorem and the Elliott–Halberstam conjecture. The innovation yielded quantitative bounds on small gaps between prime numbers and produced upper bounds on the frequency of prime tuples predicted by the Hardy–Littlewood conjecture and Prime k-tuples conjecture. The work immediately linked to ongoing efforts by researchers such as Yitang Zhang, Terence Tao, D.H.J. Polymath, Ben Green, and Tao's Polymath8 project.
Origins trace to the classic Sieve of Eratosthenes and the modern analytic developments in sieving by Atle Selberg, Heinrich Weber, and Brun. The 20th century saw advances from G.H. Hardy and J.E. Littlewood on conjectures about prime patterns, and later breakthroughs such as Bombieri–Vinogradov theorem by Enrico Bombieri and A. I. Vinogradov. The specific impetus for Maynard came after Yitang Zhang proved bounded gaps between primes in 2013, prompting rapid follow-up from Polymath Project, James Maynard, Terence Tao, and others refining the methods of Goldston–Pintz–Yıldırım and incorporating ideas from Selberg sieve theory.
The core setup chooses an admissible k-tuple of shifts connected to the Hardy–Littlewood conjecture and constructs nonnegative weight functions modeled on Selberg sieve weights. Maynard optimized linear combinations of divisor-sum weights to maximize the expected count of primes among translated integers n + h_i for i in a chosen index set. The analysis relies on distributional bounds like the Bombieri–Vinogradov theorem and conditional assumptions such as the Elliott–Halberstam conjecture or its variants to control error terms. The method expresses key quantities via convolutions involving multiplicative functions related to Möbius function and von Mangoldt function and analyzes bilinear forms appearing in the resulting quadratic forms.
Maynard proved that for any positive integer m there exist infinitely many intervals of bounded length containing at least m primes, resolving longstanding questions linked to the Prime k-tuples conjecture in a bounded-gap sense. He obtained explicit numerical bounds on gaps between consecutive primes, improving earlier results by Yitang Zhang and works in the Polymath8 project. Conditional on strong distributional hypotheses like Elliott–Halberstam conjecture, Maynard-type results recover classical conjectures such as the existence of bounded prime constellations predicted by Hardy–Littlewood conjecture. The method also led to lower bounds on the asymptotic density of prime clusters comparable to expectations from Hardy–Littlewood heuristics.
Innovations include optimized choice of sieve weights using higher-dimensional variational problems and exploitation of correlation inequalities to control variances. Maynard replaced reliance on deep Fourier-analytic dispersion estimates with flexible combinatorial optimization of weights, allowing amplification of occurrences where many of the linear forms n + h_i are prime. The proof synthesizes classical techniques from Selberg sieve with modern distributional results like Bombieri–Vinogradov theorem and ideas from Goldston–Pintz–Yıldırım method, while invoking delicate estimates for bilinear forms which relate to work by Hugh L. Montgomery, Elliott, and Vinogradov.
Immediate consequences included new unconditional bounds on prime gaps and infinitely many k-tuples with specified densities, influencing progress on the Twin Prime Conjecture landscape and stimulating refinements by researchers at Cambridge University, Harvard University, Stanford University, Columbia University, and California Institute of Technology. The technique influenced approaches to distribution problems for primes in arithmetic progressions connected to Dirichlet's theorem on arithmetic progressions, distribution of primes in short intervals studied by Atle Selberg and Montgomery–Vaughan, and to sieve-based results in additive number theory linked to Green–Tao theorem techniques. Maynard's framework also inspired algorithmic and computational investigations by groups at University College London and Imperial College London.
Open questions include sharpening quantitative bounds toward predictions of the Prime k-tuples conjecture and achieving full asymptotic formulas for prime clusters without conditional hypotheses like Elliott–Halberstam conjecture. Further work seeks to integrate stronger distributional estimates such as potential extensions of Bombieri–Vinogradov theorem or novel forms of the Generalized Riemann Hypothesis to push bounds, and to adapt Maynard-style sieves to related problems in multiplicative number theory, including patterns of prime powers and values of arithmetic functions at shifted arguments. Collaborative projects and researchers at institutions like Princeton University, Oxford University, MIT, ETH Zurich, University of Cambridge, and University of Michigan continue to explore refinements, computational verification, and cross-connections to conjectures of Hardy and Littlewood.