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Mann's Theorem
In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.Schnirelmann, L.G. (1930).On the additive properties of numbers, first published in "Proceedings of the Don Polytechnic Institute in Novocherkassk" (in Russian), vol XIV (1930), pp. 3-27, and reprinted in "Uspekhi Matematicheskikh Nauk" (in Russian), 1939, no. 6, 9–25.Schnirelmann, L.G. (1933). First published asÜber additive Eigenschaften von Zahlen in "Mathematische Annalen" (in German), vol 107 (1933), 649-690, and reprinted asOn the additive properties of numbers in "Uspekhin. Matematicheskikh Nauk" (in Russian), 1940, no. 7, 7–46. Definition The Schnirelmann density of a set of natural numbers ''A'' is defined as :\sigma A = \inf_n \frac, where ''A''(''n'') denotes the number of elements of ''A'' not exceeding ''n'' and inf is infimum.Nathanson (1996) pp.191–19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Two principal objects of study are the sumset of two subsets ''A'' and ''B'' of elements from an abelian group ''G'', :A + B = \, and the h-fold sumset of ''A'', :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of ''hA'' from the structure of ''A'': for example, determining which elements can be represented as a sum from ''hA'', where ''A'' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Mann
Henry Berthold Mann (27 October 1905, Vienna – 1 February 2000, Tucson) was a professor of mathematics and statistics at the Ohio State University. Mann proved the Schnirelmann-Landau conjecture in number theory, and as a result earned the 1946 Cole Prize. He and his student developed the ("Mann-Whitney") ''U''-statistic of nonparametric statistics. (The web-link is to a slightly updated edition of the biography.) Mann published the first mathematical book on the design of experiments: . Early life of a number theorist Born in Vienna, Austria-Hungary, to a Jewish family, Mann earned his Ph.D. degree in mathematics in 1935 from the University of Vienna under the supervision of Philipp Furtwängler. Mann immigrated to the United States in 1938, and lived in New York, supporting himself by tutoring students. In additive number theory, Mann proved the Schnirelmann–Landau conjecture on the asymptotic density of sumsets in 1942. By doing so he established Mann's theorem and earne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imre Z
Imre is a Hungarian masculine first name, which is also in Estonian use, where the corresponding name day is 10 April. It has been suggested that it relates to the name Emeric, Emmerich or Heinrich. Its English equivalents are Emery and Henry. Bearers of the name include the following (who generally held Hungarian nationality, unless otherwise noted): *Imre Antal (1935–2008), pianist *Imre Bajor (1957–2014), actor * Imre Bebek (d. 1395), baron *Imre Bródy (1891–1944), physicist * Imre Bujdosó (b. 1959), Olympic fencer *Imre Csáky (cardinal) (1672–1732), Roman Catholic cardinal * Imre Csermelyi (b. 1988), football player *Imre Cseszneky (1804–1874), agriculturist and patriot *Imre Csiszár (b. 1938), mathematician * Imre Csösz (b. 1969), Olympic judoka *Imre Czobor (1520–1581), Noble and statesman *Imre Czomba (b. 1972), Composer and musician *Imre Deme (b. 1983), football player *Imre Erdődy (1889–1973), Olympic gymnast * Imre Farkas (1879–1976), musician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eduard Wirsing
Eduard Wirsing (28 June 1931 – 22 March 2022) was a German mathematician, specializing in number theory. Biography Wirsing was born on 28 June 1931 in Berlin. Wirsing studied at the University of Göttingen and the Free University of Berlin, where he received his doctorate in 1957 under the supervision of Hans-Heinrich Ostmann with thesis ''Über wesentliche Komponenten in der additiven Zahlentheorie'' (On Essential Components in Additive Number Theory). In 1967/68 he was a professor at Cornell University and from 1969 a full professor at the University of Marburg, where he was since 1965. In 1970/71 he was at the Institute for Advanced Study. Since 1974 he was a professor at the University of Ulm, where he led the 1976 Mathematical Colloquium. He retired as professor emeritus in 1999, but continued to be mathematically active. Wirsing organized conferences on analytical number theory at the Oberwolfach Research Institute for Mathematics. In his spare time he played go and ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics () is an area of knowledge that includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (calculus and analysis). There is no general consensus about its exact scope or epistemological status. Most of mathematical activity consists of discovering and proving (by pure reasoning) properties of abstract objects. These objects are either abstractions from nature (such as natural numbers or lines), or (in modern mathematics) abstract entities of which certain properties, called axioms, are stipulated. A proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration. The result of a proof is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Khinchin
Aleksandr Yakovlevich Khinchin (russian: Алекса́ндр Я́ковлевич Хи́нчин, french: Alexandre Khintchine; July 19, 1894 – November 18, 1959) was a Soviet mathematician and one of the most significant contributors to the Soviet school of probability theory. Life and career He was born in the village of Kondrovo, Kaluga Governorate, Russian Empire. While studying at Moscow State University, he became one of the first followers of the famous Luzin school. Khinchin graduated from the university in 1916 and six years later he became a full professor there, retaining that position until his death. Khinchin's early works focused on real analysis. Later he applied methods from the metric theory of functions to problems in probability theory and number theory. He became one of the founders of modern probability theory, discovering the law of the iterated logarithm in 1924, achieving important results in the field of limit theorems, giving a definition of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harald Helfgott
Harald Andrés Helfgott (born 25 November 1977) is a Peruvian mathematician working in number theory. Helfgott is a researcher ('' directeur de recherche'') at the CNRS at the Institut Mathématique de Jussieu, Paris. Early life and education Helfgott was born on 25 November 1977 in Lima, Peru. He graduated from Brandeis University in 1998 ( BA, summa cum laude). He received his Ph.D. from Princeton University in 2003 under the direction of Henryk Iwaniec and Peter Sarnak, with the thesis ''Root numbers and the parity problem''. Career Helfgott was a post-doctoral Gibbs Assistant Professor at Yale University from 2003 to 2004. He was then a post-doctoral fellow at CRM–ISM–Université de Montréal from 2004 to 2006. Helfgott was a Lecturer, Senior Lecturer, and then Reader at the University of Bristol from 2006 to 2011. He has been a researcher at the CNRS since 2010, initially as a ''chargé de recherche première classe'' at the École normale supérieure before becomin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Charles Vaughan (mathematician)
Robert Charles "Bob" Vaughan FRS (born 24 March 1945) is a British mathematician, working in the field of analytic number theory. Life Since 1999 he has been Professor at Pennsylvania State University, and since 1990 Fellow of the Royal Society. He did his PhD at the University of London under supervision of Theodor Estermann. He supervised Trevor Wooley's PhD. Awards In 2012 he became a fellow of the American Mathematical Society. retrieved 2013-08-28. See also *Vaughan's identity
In mathematics and analytic number theory, Vaughan's identity is an identity found by that can be used ...
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Hans Riesel
Hans Ivar Riesel (May 28, 1929 in Stockholm – December 21, 2014) was a Swedish mathematician who discovered the 18th known Mersenne prime in 1957, using the computer BESK: this prime is 23217-1 and consists of 969 digits. He held the record for the largest known prime from 1957 to 1961, when Alexander Hurwitz discovered a larger one. Riesel also discovered the Riesel numbers as well as developing the Lucas–Lehmer–Riesel test. After having worked at the Swedish Board for Computing Machinery, he was awarded his Ph.D. from Stockholm University in 1969 for his thesis ''Contributions to numerical number theory'',LIBRISbr>1768091/ref> and in the same year joined the Royal Institute of Technology as a senior lecturer and associate professor. Selected publications * See also * Riesel number * Riesel Sieve Riesel Sieve was a volunteer computing project, running in part on the BOINC platform. Its aim was to prove that 509,203 is the smallest Riesel number, by finding a prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olivier Ramaré
Olivier Ramaré is a French mathematician who works as Senior researcher for the CNRS. He is currently attached to Aix-Marseille Université. Ramaré earned a doctorate in 1991 from the University of Bordeaux with a dissertation ''Contribution au problème de Goldbach : tout entier >1 est somme d'au plus treize nombres premiers'' supervised by Jean-Marc Deshouillers. In 1995, he sharpened earlier work on Schnirelmann's theorem by proving that every even number is a sum of at most six primes. This result may be compared with Goldbach's conjecture, which states that every even number except 2 is the sum of two primes. The truth of Ramaré's result for all sufficiently large even numbers is a consequence of Vinogradov's theorem, whereas the full result follows from Goldbach's weak conjecture. In turn, Ramaré's result was strengthened by Terence Tao who proved in 2014 that every odd number is the sum of at most five primes, and by Harald Helfgott's claimed proof of Goldbach's wea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
