|
Hardy–Ramanujan Theorem
In mathematics, the Hardy–Ramanujan theorem, proved by Ramanujan and checked by Hardy states that the normal order of the number \omega(n) of distinct prime factors of a number n is \log\log n. Roughly speaking, this means that most numbers have about this number of distinct prime factors. Precise statement A more precise version states that for every real-valued function \psi(n) that tends to infinity as n tends to infinity , \omega(n)-\log\log n, <\psi(n)\sqrt or more traditionally for '''' (all but an proportion of) integers. That is, let be the number of positive integers less than for which the abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Kac Theorem
In number theory, the Erdős–Kac theorem, named after Paul Erdős and Mark Kac, and also known as the fundamental theorem of probabilistic number theory, states that if ''ω''(''n'') is the number of distinct prime factors of ''n'', then, loosely speaking, the probability distribution of : \frac is the standard normal distribution. (\omega(n) is sequence OEIS:A001221, A001221 in the On-Line Encyclopedia of Integer Sequences, OEIS.) This is an extension of the Hardy–Ramanujan theorem, which states that the Normal order of an arithmetic function, normal order of ''ω''(''n'') is log log ''n'' with a typical error of size \sqrt. Precise statement For any fixed ''a'' < ''b'', : where is the normal (or "Gaussian") distribution, defined as : More generally, if ''f''(''n'') is a strongly additive function ( |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán–Kubilius Inequality
The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory. It is useful for proving results about the normal order of an arithmetic function. The theorem was proved in a special case in 1934 by Pál Turán and generalized in 1956 and 1964 by Jonas Kubilius. Statement of the theorem This formulation is from Tenenbaum. Other formulations are in Narkiewicz and in Cojocaru & Murty. Suppose ''f'' is an additive complex-valued arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ..., and write ''p'' for an arbitrary prime and for an arbitrary positive integer. Write :A(x)=\sum_ f(p^\nu) p^(1-p^) and :B(x)^2 = \sum_ \left, f(p^\nu) \ ^2 p^. Then there is a function ε(''x'') that goes to zero when ''x'' goes to infinity, and such that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Prime
In number theory, a natural number is called -almost prime if it has prime factors. More formally, a number is -almost prime if and only if , where is the total number of primes in the prime factorization of (can be also seen as the sum of all the primes' exponents): :\Omega(n) := \sum a_i \qquad\mbox\qquad n = \prod p_i^. A natural number is thus prime if and only if it is 1-almost prime, and semiprime if and only if it is 2-almost prime. The set of -almost primes is usually denoted by . The smallest -almost prime is . The first few -almost primes are: : The number of positive integers less than or equal to with exactly prime divisors (not necessarily distinct) is asymptotic to: : \pi_k(n) \sim \left( \frac \right) \frac, a result of Landau Landau (), officially Landau in der Pfalz (, ), is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, titled ''Proceedings of the Cambridge Philosophical Society'' before 1975, has been published since 1843. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website References Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stein's Method
Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972, to obtain a bound between the distribution of a sum of m-dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform) metric and hence to prove not only a central limit theorem, but also bounds on the rates of convergence for the given metric. History At the end of the 1960s, unsatisfied with the by-then known proofs of a specific central limit theorem, Charles Stein developed a new way of proving the theorem for his statistics lecture. . Interview given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying exceptions (for example, ''double roots'' counted twice). Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of ''distinct'' elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "distinct". Multiplicity of a prime factor In prime factorization, the multiplicity of a prime factor is its p-adic valuation. For example, the prime factorization of the integer is : the multiplicity of the prime factor is , while the multiplicity of each of the prime factors and is . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turán Sieve
In number theory, the Turán 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 Pál Turán in 1934. Description In terms of sieve theory the Turán sieve is of ''combinatorial type'': deriving from a rudimentary form of the inclusion–exclusion principle. The result gives an ''upper bound'' for the size of the sifted set. Let ''A'' be a set of positive integers ≤ ''x'' and let ''P'' be a set of primes. For each ''p'' in ''P'', let ''A''''p'' denote the set of elements of ''A'' divisible by ''p'' and extend this to let ''A''''d'' be the intersection of the ''A''''p'' for ''p'' dividing ''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 ≤ ''z''. The object of the sieve is to estimate :S(A,P,z) = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pál Turán
Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. In 1940, because of his Jewish origins, he was arrested by History of the Jews in Hungary#The Holocaust, the Nazis and sent to a Labour service (Hungary), labour camp in Transylvania, later being transferred several times to other camps. While imprisoned, Turán came up with some of his best theories, which he was able to publish after the war. Turán had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. Biography Early years Turán was born into a Jews of Hungary, Hungarian Jewish family in Budapest on 18 August 1910. Pál's outstanding mathematical abilities showed early, already in secondary school he was the best student. At the same period of time, Turán and Pál Erdős were famous answerers in the journal ''KöMaL''. On 1 September 1930, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
