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Jurkat–Richert Theorem
The Jurkat–Richert theorem is a mathematical theorem in sieve theory. It is a key ingredient in proofs of Chen's theorem on Goldbach's conjecture. It was proved in 1965 by Wolfgang B. Jurkat and Hans-Egon Richert. Statement of the theorem This formulation is from Diamond & Halberstam. Other formulations are in Jurkat & Richert, Halberstam & Richert, and Nathanson. Suppose ''A'' is a finite sequence of integers and ''P'' is a set of primes. Write ''A''''d'' for the number of items in ''A'' that are divisible by ''d'', and write ''P''(''z'') for the product of the elements in ''P'' that are less than ''z''. Write ω(''d'') for a 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'') i ... such that ω(''p'')/''p'' is approximately the proportion of elements of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...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] |
Chen's Theorem
In number theory, Chen's theorem states that every sufficiently large parity (mathematics), even number can be written as the sum of either two prime number, primes, or a prime and a semiprime (the product of two primes). History The theorem was first stated by China, Chinese mathematician Chen Jingrun in 1966, with further details of the mathematical proof, proof in 1973. His original proof was much simplified by P. M. Ross in 1975. Chen's theorem is a giant step towards the Goldbach's conjecture, and a remarkable result of the sieve theory, sieve methods. Chen's theorem represents the strengthening of a previous result due to Alfréd Rényi, who in 1947 had shown there exists a finite ''K'' such that any even number can be written as the sum of a prime number and the product of at most ''K'' primes. Variations Chen's 1973 paper stated two results with nearly identical proofs. His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the Twi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldbach's Conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort. History On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would indeed be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...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] |
Acta Arithmetica
''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences The Institute of Mathematics of the Polish Academy of Sciences is a research institute of the Polish Academy of Sciences.Online archives (Library of Science, Issues: 1935–2000) 1935 establishments in Poland [...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] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge
Cambridge ( ) is a university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge became an important trading centre during the Roman and Viking ages, and there is archaeological evidence of settlement in the area as early as the Bronze Age. The first town charters were granted in the 12th century, although modern city status was not officially conferred until 1951. The city is most famous as the home of the University of Cambridge, which was founded in 1209 and consistently ranks among the best universities in the world. The buildings of the university include King's College Chapel, Cavendish Laboratory, and the Cambridge University Library, one of the largest legal deposit libraries in the world. The city's skyline is dominated by several college buildings, along with the spire of the Our Lady and the English Martyrs ... [...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] |
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