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| Brun sieve | |
|---|---|
| Name | Brun sieve |
| Mathematician | Viggo Brun |
| Field | Number theory |
| Introduced | 1915 |
| Concepts | Sieve methods, twin primes, Brun's theorem, upper bound sieves |
Brun sieve The Brun sieve is an analytic number theory sieve method introduced to study prime number patterns such as twin primes, leading to Brun's theorem on the convergence of reciprocal sums of twin primes. It provided the first nontrivial finite bounds separating prime pairs and influenced later developments like the Selberg sieve and the Large sieve. The method connects with works by Legendre, Eratosthenes, Legendre's formula, Viggo Brun, G. H. Hardy, John Edensor Littlewood, and Atle Selberg.
Brun's work originated in early 20th-century Scandinavian mathematics, building on classical ideas from Eratosthenes and the combinatorial sieves of Édouard Lucas and Adrien-Marie Legendre. Viggo Brun formulated a combinatorial apparatus that yielded explicit upper bounds for counting integers free of small prime factors, addressing conjectures such as the twin prime conjecture and questions raised by Goldbach conjecture investigations. The approach is contemporaneous with analytic contributions by G. H. Hardy and John Edensor Littlewood and presaged techniques later refined by Atle Selberg and Harald Cramér.
Brun published his main results in 1915 after earlier contributions around 1910, amid research by Jacques Hadamard, Charles-Jean de la Vallée Poussin, and investigators of prime distribution in France and Britain. The sieve addressed limitations from Legendre and from early 19th-century work by Joseph-Louis Lagrange and Carl Friedrich Gauss on prime counting. Following Brun, researchers including Alfréd Rényi, Paul Erdős, Atle Selberg, Heini Halberstam, and Hans-Egon Richert extended sieve theory, while computational studies by Daniel Shanks and later by John Brillhart connected theory to explicit prime tables. The method stimulated international collaboration across Norway, Hungary, United States, and United Kingdom mathematical communities.
Brun produced bounds for the counting function of integers n ≤ x for which a fixed set of linear forms produce primes, notably for pairs (n, n+2); he proved that the sum of reciprocals of twin primes converges (Brun's theorem). The sieve gives an upper bound of the form S(x, z) ≤ V(z) x + error terms, where V(z) is a multiplicative factor determined by primes up to z; this structure connects to results by Rosser and Schoenfeld on explicit inequalities and to abstract frameworks by Heini Halberstam and H. E. Richert. Brun's theorem contrasts with the divergent harmonic series of all primes proved by Leonhard Euler and is a milestone preceding conditional advances like the Bombieri–Vinogradov theorem and the Green–Tao theorem.
Brun's method employs combinatorial inclusion–exclusion truncated at a finite level and weights to control main and error terms, resembling later weighted sieves studied by Atle Selberg and Alfréd Rényi. One constructs upper-bound linear forms and applies multiplicative function estimates influenced by Dirichlet and Bernhard Riemann techniques, invoking properties of the Möbius function and partial summation used in works by G. H. Hardy and John Edensor Littlewood. The proof balances contributions from small and large primes, introduces Brun's weights to optimize cancellations, and derives convergence of certain additive series via explicit estimates analogous to bounds used in the Prime Number Theorem proofs by Hadamard and de la Vallée Poussin.
The Brun sieve yielded concrete upper bounds for prime tuples and produced Brun's theorem on twin primes, impacting research by Paul Erdős on additive number theory and prompting the formulation of the parity problem in sieve theory addressed by Henryk Iwaniec and Enrico Bombieri. It influenced bounds in results by Chen Jingrun on primes plus almost-primes, and informed computational verification programs by Atkin and Morain for primality. Later methods like the Selberg sieve, large sieve, and the combinatorial sieve trace conceptual lineage to Brun; consequences include tools used in the proofs of bounded gaps between primes by researchers such as Yitang Zhang, James Maynard, and teams led by Terence Tao.
Variants include the weighted Brun sieve, multidimensional Brun-type sieves for prime k-tuples developed by Halberstam and Richert, and hybrid sieves blending Brun ideas with analytic estimates found in the Bombieri–Vinogradov theorem. Generalizations led to the Rosser–Iwaniec sieve, the Selberg sieve, and modern formulations in the sieve of Eratosthenes framework adapted in computational algebra systems used by Richard Brent and John Brillhart. Extensions handle polynomial sequences and linear forms as in work by Bunyakovsky and investigations influenced by Hardy–Littlewood conjectures.
Classical applications compute Brun's constant B2, the sum of reciprocals of twin primes, approximated numerically by teams including Thomas R. Nicely, Nicely's collaborators, and computational projects at institutions like Oak Ridge National Laboratory and Los Alamos National Laboratory. Explicit sieve counts for x up to large bounds rely on combinatorial sieves combined with prime tables compiled by Nicely, Daniel Shanks, and D. H. Lehmer. Worked examples show upper bounds for twin primes ≤ x of order x / (log x)^2 with explicit constants, paralleling numerical verifications in datasets curated by Oliveira e Silva and used in heuristics related to the Hardy–Littlewood k-tuple conjecture.
Category:Sieve methods