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Linnik's Theorem
Linnik's theorem in analytic number theory answers a natural question after Dirichlet's theorem on arithmetic progressions. It asserts that there exist positive ''c'' and ''L'' such that, if we denote p(''a'',''d'') the least primes in arithmetic progression, prime in the arithmetic progression :a + nd,\ where ''n'' runs through the positive integers and ''a'' and ''d'' are any given positive coprime integers with 1 ≤ ''a'' ≤ ''d'' − 1, then: : \operatorname(a,d) < c d^. \; The theorem is named after Yuri Vladimirovich Linnik, who mathematical proof, proved it in 1944. Although Linnik's proof showed ''c'' and ''L'' to be effective results in number theory, effectively computable, he provided no numerical values for them. It follows from Zsigmondy's theorem that p(1,''d'') ≤ 2''d'' − 1, for all ''d'' ≥ 3. It is known that p(1,''p'') ≤ L''p'', for all prime number, primes ''p'' ≥ 5, as L''p'' is modular arithmetic, congruent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * * * * ISI Ale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proc
Proc may refer to: * Proč, a village in eastern Slovakia * '' Proč?'', a 1987 Czech film * procfs or proc filesystem, a special file system (typically mounted to ) in Unix-like operating systems for accessing process information * Protein C (PROC) * Proc, a term in video game terminology * Procedures or process, in the programming language ALGOL 68 * People's Republic of China, the formal name of China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's most populous country, with a population exceeding 1.4 billion, slightly ahead of India. China spans the equivalent of five time zones and ... * the official acronym for the Canadian House of Commons Standing Committee on Procedure and House Affairs {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Heath-Brown
David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory. Education He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker. Career and research In 1979 he moved to the University of Oxford, where from 1999 he held a professorship in pure mathematics. He retired in 2016. Heath-Brown is known for many striking results. He proved that there are infinitely many prime numbers of the form ''x''3 + 2''y''3. In collaboration with S. J. Patterson in 1978 he proved the Kummer conjecture on cubic Gauss sums in its equidistribution form. He has applied Burgess's method on character sums to the ranks of elliptic curves in families. He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0. Heath-Brown also showed that Linnik's constant is less than or equal to 5.5. More recently, Heath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liu Jian Min
/ ( or ) is an East Asian surname. pinyin: in Mandarin Chinese, in Cantonese. It is the family name of the Han dynasty emperors. The character originally meant 'kill', but is now used only as a surname. It is listed 252nd in the classic text Hundred Family Surnames. Today, it is the 4th most common surname in Mainland China as well as one of the most common surnames in the world. Distribution In 2019 劉 was the fourth most common surname in Mainland China. Additionally, it was the most common surname in Jiangxi province. In 2013 it was found to be the 5th most common surname, shared by 67,700,000 people or 5.1% of the population, with the province with the most people being Shandong.中国四百大姓, 袁义达, 邱家儒, Beijing Book Co. Inc., 1 January 2013 Origin One source is that they descend from the Qí (祁) clan of Emperor Yao. For example the founding emperor of the Han dynasty (one of China's golden ages), Liu Bang (Emperor Gaozu of Han) was a descendant of E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Arith
''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] |
Sidney Graham
Sidney West Graham is a mathematician interested in analytic number theory and professor at Central Michigan University. He received his Ph.D., which was supervised by Hugh Montgomery, from the University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ... in 1977. In his Ph.D. thesis he lowered the upper bound for Linnik's constant to 36 and subsequently reduced the bound further to 20.. References External linksSidney West Graham page at Central Michigan University 20th-century American mathematicians 21st-century American mathematicians Number theorists University of Michigan alumni Living people Central Michigan University faculty 1950 births {{US-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matti Jutila
Matti Ilmari Jutila (born 1943) is a mathematician and a professor emeritus at the University of Turku. He researches in the field of analytic number theory. Education and career Jutila completed a doctorate at the University of Turku in 1970, with a dissertation related to Linnik's constant supervised by . Jutila's work has repeatedly succeeded in lowering the upper bound for Linnik's constant. He is the author of a monograph, ''Lectures on a method in the theory of exponential sums'' (1987). He has been a member of the Finnish Academy of Science and Letters The Finnish Academy of Science and Letters (Finnish ''Suomalainen Tiedeakatemia''; Latin ''Academia Scientiarum Fennica'') is a Finnish learned society. It was founded in 1908 and is thus the second oldest academy in Finland. The oldest is the Fi ... since 1982. References External links * 1943 births Living people Finnish mathematicians Number theorists Members of the Finnish Academy of Science and Letters< ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chen Jingrun
Chen Jingrun (; 22 May 1933 – 19 March 1996), also known as Jing-Run Chen, was a Chinese mathematician who made significant contributions to number theory, including Chen's theorem and the Chen prime. Life and career Chen was the third son in a large family from Fuzhou, Fujian, China. His father was a postal worker. Chen Jingrun graduated from the Mathematics Department of Xiamen University in 1953. His advisor at the Chinese Academy of Sciences was Hua Luogeng. His work on the twin prime conjecture, Waring's problem, Goldbach's conjecture and Legendre's conjecture led to progress in analytic number theory. In a 1966 paper he proved what is now called Chen's theorem: every sufficiently large even number can be written as the sum of a prime and a semiprime (the product of two primes) – e.g., 100 = 23 + 7·11. Despite being persecuted during the Cultural Revolution, he expanded his proof in the 1970s. After the end of the Cultural Revolution, Xu Chi wro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pan Chengdong
Pan Chengdong ( zh, c=潘承洞, p=Pān Chéngdòng; 26 May 1934 – 27 December 1997) was a Chinese mathematician who made numerous contributions to number theory, including progress on Goldbach's conjecture. He was vice president of Shandong University and took the role of president from 1986 to 1997. Born in Suzhou, Jiangsu Province on 26 May 1934, he entered the Department of Mathematics and Mechanics of Peking University in 1952 and obtained a postgraduate degree in 1961 advised by Min Sihe, a student of Edward Charles Titchmarsh Edward Charles "Ted" Titchmarsh (June 1, 1899 – January 18, 1963) was a leading British mathematician. Education Titchmarsh was educated at King Edward VII School (Sheffield) and Balliol College, Oxford, where he began his studies in October .... He then went to work at the Department of Mathematics of Shandong University. He was honored with an Academician of the Chinese Academy of Science in 1991. Previously, Wang Yuan made progress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
