Selberg sieve

Selberg sieve
Konrad Jacobs, Erlangen · CC BY-SA 2.0 de · source
Name	Selberg sieve
Field	Analytic number theory
Introduced	1947
Inventor	Atle Selberg
Applications	Prime number theory, twin primes problem, Goldbach conjecture

Selberg sieve The Selberg sieve is a combinatorial analytic method in analytic number theory devised by Atle Selberg to estimate the size of sifted sets such as almost-primes and integers free of small prime factors. It provides systematic construction of nonnegative weight functions and upper bounds for counting functions that arise in problems connected to the prime number theorem, the twin prime conjecture, and additive problems like the Goldbach conjecture. The method interacts with ideas from Paul Erdős, G. H. Hardy, and John Littlewood, and has been adapted by figures such as Daniel Goldston, János Pintz, and Yitang Zhang.

Introduction

The Selberg sieve sits alongside the Legendre sieve, the Brun sieve, and the Combinatorial sieve as a central tool in sieve theory. It systematically builds upper-bound sieves using quadratic forms in arithmetic functions inspired by Selberg's work in the late 1940s, following influences from Srinivasa Ramanujan and Ivan Vinogradov. The approach yields explicit, often sharp, bounds for sifted sets relevant to the distribution of primes, studies of smooth numbers, and investigations into patterns like prime gaps and prime tuples conjecture.

History and development

Selberg introduced his sieve in a sequence of papers and lectures motivated by problems posed by Paul Turán and Atle Selberg's contemporaries in Norway and Sweden. Early recipients of Selberg's ideas included Heini Halberstam and Hans-Egon Richert, who incorporated the method into the textbook development of sieve methods. During the mid-20th century, collaborations and rivalries among Pál Erdős, János Magyar-era mathematicians, and the British mathematical community fostered refinements that connected Selberg's weights to work by Vaughan and Rosser. Later developments by A. G. Postnikov, D. A. Burgess, and modern contributors like Goldston, Pintz, and Yıldırım integrated Selberg-type sieves into breakthroughs on small gaps between primes and bounded gap results.

Statement and setup

The classical Selberg sieve considers a finite integer sequence A contained in an interval up to X and a set of primes P up to some bound z, producing the sifted set S(A,P,z) of elements of A coprime to the product of primes in P. One introduces multiplicative functions and sieve weights λ_d supported on divisors composed of primes in P to estimate the counting function S(A,P,z). The aim is to minimize the quadratic form sum_{d,e} λ_d λ_e R([d,e]) subject to normalization constraints, where R(n) denotes the remainder term connected to arithmetic progressions modulo n, drawing on results from Dirichlet, Chebyshev, and the Bombieri–Vinogradov theorem in controlling error terms.

Fundamental lemmas and weights

Central to the method is the construction of nonnegative weights λ_d via optimization of a bilinear or quadratic form, often achieved by solving a finite-dimensional least-squares problem. The Selberg sieve upper bound lemma gives S(A,P,z) ≤ ∑_{d|P(z)} λ_d A_d, with the choice of λ_d minimizing the variance; here A_d denotes the count of elements divisible by d. The fundamental lemma relates the main term to multiplicative functions like the Möbius function and the Euler phi function and uses asymptotic estimates linked to the prime number theorem and zero-free regions for Riemann zeta function in refined analyses employed by Iwaniec and Friedlander.

Applications and examples

Selberg's method yields quantitative bounds for the number of integers free of small prime factors, estimates for almost-primes (numbers with few prime factors) used in results by Chen Jingrun on Goldbach's weak conjecture variants, and upper bounds in the twin primes problem as pursued by Brun and later by Chen. It underpins sieve weights in proofs of bounded gaps between primes by teams including Goldston–Pintz–Yıldırım and influenced the work of Zhang. Selberg-type sieves also feature in results on primes in arithmetic progressions related to the Bombieri–Vinogradov theorem and in combinatorial applications studied by Granville and Soundararajan.

Variants and generalizations

Variants include the linear Selberg sieve, the combinatorial Selberg sieve, and bilinear forms that combine Selberg weights with dispersion methods from Linnik and Duke. Generalizations integrate the large sieve inequalities associated with Elliott and Gallagher, the dispersion method of Fouvry and Iwaniec, and adaptations to trace functions and automorphic forms studied by Kowalski and Michel. Hybrid sieves combine Selberg weights with Maynard-style multidimensional sieves to produce sharper results on prime clusters, with contributions from Maynard and Tao.

Proofs and key techniques

Proofs center on quadratic optimization over divisor-supported coefficients and manipulation of multiplicative convolutions, employing estimates from Dirichlet L-functions, zero-density results for the Riemann zeta function, and bilinear sum bounds akin to methods by Bombieri and Iwaniec. Key technical tools include the Möbius inversion principle associated historically with Augustin-Louis Cauchy-era combinatorial identities, the saddle-point heuristic found in asymptotic analysis by E. T. Whittaker-influenced authors, and spectral techniques when Selberg weights interact with automorphic forms in the work of Selberg's associates. Modern proofs often combine harmonic analysis on adelic groups and exponential sum estimates pioneered by Weyl and Vinogradov.

Category:Analytic number theory