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Fundamental Lemma Of Sieve Theory
In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems. Halberstam & Richert write: Diamond & Halberstam attribute the terminology ''Fundamental Lemma'' to Jonas Kubilius. Common notation We use these notations: * A is a set of X positive integers, and A_d is its subset of integers divisible by d * w(d) and R_d are functions of A and of d that estimate the number of elements of A that are divisible by d, according to the formula : \left\vert A_d \right\vert = \frac X + R_d . :Thus w(d)/d represents an approximate density of members divisible by ''d'', and R_d represents an error or remainder term. * P is a set of primes, and P(z) is the product of those primes \leq z * S(A, P, z) is the number of elements of A not divisible by any prime in P that is \leq z * \kappa is a constant, called the sifting density, that appears in the assumptions below. It is a weighted average ... [...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] |
Heini Halberstam
Heini Halberstam (11 September 1926[Doreen Halberstam, wife] – 25 January 2014) was a Czech-born British mathematician, working in the field of analytic number theory. He is remembered in part for the Elliott–Halberstam conjecture from 1968. Life and career Halberstam was born in Most (Most District), Most, Czechoslovakia and died in Champaign, Illinois, US. His father died when he was very young. After Adolf Hitler's annexation of the Sudetenland, he and his mother moved to Prague. At the age of twelve, as the Nazi occupation progressed, he was one of the 669 children saved by Nicholas Winton, Sir Nicholas Winton, who organized the Kindertransport, a train that allowed those children to leave Nazi-occupied territory. He was sent to England, where he lived during World War II, World War II. He obtained his PhD in 1952, from University College London, University College, London, under supervision of Theodor Estermann. From 1962 until 1964, Halberstam was Erasmus Smith's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans-Egon Richert
Hans-Egon Richert (June 2, 1924 – November 25, 1993) was a German mathematician who worked primarily in analytic number theory. He is the author (with Heini Halberstam) of a definitive book on sieve theory. Life and education Hans-Egon Richert was born in 1924 in Hamburg, Germany. He attended the University of Hamburg and received his Ph.D under Max Deuring in 1950. He held a temporary chair at the University of Göttingen and then a newly created chair at the University of Marburg. In 1972 he moved to the University of Ulm, where he remained until his retirement in 1991. He died on November 25, 1993 in Blaustein, near Ulm, Germany. Work Richert worked primarily in analytic number theory, and beginning around 1965 started a collaboration with Heini Halberstam and shifted his focus to sieve theory. For many years he was a chairman of the Analytic Number Theory meetings at the Mathematical Research Institute of Oberwolfach. Analytic number theory Richert made contribution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brun Sieve
In the field of number theory, the Brun sieve (also called Brun's pure 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 Viggo Brun in 1915 and later generalized to the fundamental lemma of sieve theory by others. Description In terms of sieve theory the Brun sieve is of ''combinatorial type''; that is, it derives from a careful use of the inclusion–exclusion principle. Let A be a finite set of positive integers. Let P be some set of prime numbers. For each prime p in P, let A_p denote the set of elements of A that are divisible by p. This notation can be extended to other integers d that are products of distinct primes in P. In this case, define A_d to be the intersection of the sets A_p for the prime factors p of d. Finally, define A_1 to be A itself. Let z be an arbitrary positive real number. The object of the sieve is to estimate: S(A,P,z) = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jonas Kubilius
Jonas Kubilius (27 July 1921 – 30 October 2011) was a Lithuanian mathematician who worked in probability theory and number theory. He was rector of Vilnius University for 32 years, and served one term in the Lithuanian parliament. Life and education Kubilius was born in Fermos village, Eržvilkas county, Jurbarkas district municipality, Lithuania on 27 July 1921. He graduated from Raseiniai high school in 1940 and entered Vilnius University, from which he graduated '' summa cum laude'' in 1946 after taking off a year to teach mathematics in middle school. Kubilius received the Candidate of Sciences degree in 1951 from Leningrad University. His thesis, written under Yuri Linnik, was titled ''Geometry of Prime Numbers''. He received the Doctor of Science degree ( habilitation) in 1957 from the Steklov Institute of Mathematics in Moscow. Career Kubilius had simultaneous careers at Vilnius University and at the Lithuanian Academy of Sciences. He continued working at the unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighted Average
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox. Examples Basic example Given two school with 20 students, one with 30 test grades in each class as follows: :Morning class = :Afternoon class = The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Class
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Friedlander
John Friedlander is a Canadian mathematician specializing in analytic number theory. He received his B.Sc. from the University of Toronto in 1965, an M.A. from the University of Waterloo in 1966, and a Ph.D. from Pennsylvania State University in 1972. He was a lecturer at M.I.T. in 1974–76, and has been on the faculty of the University of Toronto since 1977, where he served as Chair during 1987–91. He has also spent several years at the Institute for Advanced Study. In addition to his individual work, he has been notable for his collaborations with other well-known number theorists, including Enrico Bombieri, William Duke, Andrew Granville, and especially Henryk Iwaniec. In 1997, in joint work with Henryk Iwaniec, Friedlander proved that infinitely many prime numbers can be obtained as the sum of a square and fourth power: . Friedlander and Iwaniec improved Enrico Bombieri's "asymptotic sieve" technique to construct their proof. Awards and honors In 1999, Friedlander receiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henryk Iwaniec
Henryk Iwaniec (born October 9, 1947) is a Polish-American mathematician, and since 1987 a professor at Rutgers University. Background and education Iwaniec studied at the University of Warsaw, where he got his PhD in 1972 under Andrzej Schinzel. He then held positions at the Institute of Mathematics of the Polish Academy of Sciences until 1983 when he left Poland. He held visiting positions at the Institute for Advanced Study, University of Michigan, and University of Colorado Boulder before being appointed Professor of Mathematics at Rutgers University. He is a citizen of both Poland and the United States. He and mathematician Tadeusz Iwaniec are twin brothers. Work Iwaniec studies both sieve methods and deep complex-analytic techniques, with an emphasis on the theory of automorphic forms and harmonic analysis. In 1997, Iwaniec and John Friedlander proved that there are infinitely many prime numbers of the form . Results of this strength had previously been seen as co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime. An arithmetic function ''f''(''n'') is said to be completely multiplicative (or totally multiplicative) if ''f''(1) = 1 and ''f''(''ab'') = ''f''(''a'')''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not coprime. Examples Some multiplicative functions are defined to make formulas easier to write: * 1(''n''): the constant function, defined by 1(''n'') = 1 (completely multiplicative) * Id(''n''): identity function, defined by Id(''n'') = ''n'' (completely multiplicative) * Id''k''(''n''): the power functions, defined by Id''k''(''n'') = ''n''''k'' for any complex number ''k'' (completely multiplicative). As special cases we have ** Id0(''n'') = 1(''n'') and ** Id1(''n'') = Id(''n''). * ''ε''(''n''): the function defined by ''ε''(''n'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inclusion–exclusion Principle
In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as : , A \cup B, = , A, + , B, - , A \cap B, where ''A'' and ''B'' are two finite sets and , ''S'', indicates the cardinality of a set ''S'' (which may be considered as the number of elements of the set, if the set is finite). The formula expresses the fact that the sum of the sizes of the two sets may be too large since some elements may be counted twice. The double-counted elements are those in the intersection of the two sets and the count is corrected by subtracting the size of the intersection. The inclusion-exclusion principle, being a generalization of the two-set case, is perhaps more clearly seen in the case of three sets, which for the sets ''A'', ''B'' and ''C'' is given by :, A \cup B \cup C, = , A, + , B, + , C, - , A \cap B, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
