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Larger Sieve
In number theory, the larger sieve is a sieve invented by Patrick X. Gallagher. The name denotes a heightening of the large sieve. Combinatorial sieves like the Selberg sieve are strongest, when only a few residue classes are removed, while the term large sieve means that this sieve can take advantage of the removal of a large number of up to half of all residue classes. The larger sieve can exploit the deletion of an arbitrary number of classes. Statement Suppose that \mathcal is a set of prime powers, ''N'' an integer, \mathcal a set of integers in the interval , ''N'' such that for q\in \mathcal there are at most g(q) residue classes modulo q, which contain elements of \mathcal. Then we have :, \mathcal, \leq \frac, provided the denominator on the right is positive. Applications A typical application is the following result, for which the large sieve fails (specifically for \theta>\frac), due to Gallagher: The number of integers n\leq N, such that the order of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sieve Theory
Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit ''X''. Correspondingly, the prototypical example of a sieve is the sieve of Eratosthenes, or the more general Legendre sieve. The direct attack on prime numbers using these methods soon reaches apparently insuperable obstacles, in the way of the accumulation of error terms. In one of the major strands of number theory in the twentieth century, ways were found of avoiding some of the difficulties of a frontal attack with a naive idea of what sieving should be. One successful approach is to approximate a specific sifted set of numbers (e.g. the set of prime numbers) by another, simpler set (e.g. the set of almost prime numbers), which is typically somewhat larger than the original set, and easier to analyze. More sophisticated sieves als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Patrick X
Patrick may refer to: * Patrick (given name), list of people and fictional characters with this name * Patrick (surname), list of people with this name People * Saint Patrick (c. 385–c. 461), Christian saint *Gilla Pátraic (died 1084), Patrick or Patricius, Bishop of Dublin * Patrick, 1st Earl of Salisbury (c. 1122–1168), Anglo-Norman nobleman * Patrick (footballer, born 1983), Brazilian right-back * Patrick (footballer, born 1985), Brazilian striker *Patrick (footballer, born 1992), Brazilian midfielder * Patrick (footballer, born 1994), Brazilian right-back *Patrick (footballer, born May 1998), Brazilian forward *Patrick (footballer, born November 1998), Brazilian attacking midfielder * Patrick (footballer, born 1999), Brazilian defender * Patrick (footballer, born 2000), Brazilian defender *John Byrne (Scottish playwright) (born 1940), also a painter under the pseudonym Patrick *Don Harris (wrestler) (born 1960), American professional wrestler who uses the ring name Patrick ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Large Sieve
The large sieve is a method (or family of methods and related ideas) in analytic number theory. It is a type of sieve where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein only a few residue classes are removed. The method has been further heightened by the larger sieve which removes arbitrarily many residue classes. Name Its name comes from its original application: given a set S \subset \ such that the elements of ''S'' are forbidden to lie in a set ''Ap'' ⊂ Z/''p'' Z modulo every prime ''p'', how large can ''S'' be? Here ''A''''p'' is thought of as being large, i.e., at least as large as a constant times ''p''; if this is not the case, we speak of a ''small sieve''. History The early history of the large sieve traces back to work of Yu. B. Linnik, in 1941, working on the problem of the least quadratic non-residue. Subsequently Alfréd Rényi worked on it, using probability methods. It was only two deca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg Sieve
In number theory, the Selberg sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s. Description In terms of sieve theory the Selberg sieve is of ''combinatorial type'': that is, derives from a careful use of the inclusion–exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an ''upper bound'' for the size of the sifted set. Let A be a set of positive integers \le x and let P be a set of primes. Let A_d denote the set of elements of A divisible by d when d is a product of distinct primes from P. Further let A_1 denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are \le z. The object of the sieve is to estimate :S(A,P,z) = \left\vert A \setminus \bigc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathematics related to work by Hungarian mathematicians. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.39, and its 2015 impact factor was 0.469. The editor-in-chief is Imre Bárány, honorary editor is Ákos Császár, the editors are the mathematician members of the Hungarian Academy of Sciences. Abstracting and indexing According to the ''Journal Citation Reports'', the journal had a 2020 impact factor of 0.623. This journal is indexed by the following services: * Science Citation Index * Journal Citation Reports/Science Edition * Scopus * Mathematical Reviews * Zentralblatt Math zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
