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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, 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 for ''x'' ≥ 2 we have :\frac \sum_ , f(n) - A(x), ^2 \le (2 + \varepsilon(x)) B(x)^2. Applications of the theorem Turán developed the inequality to create a simpler proof of the Hardy–Ramanujan theorem about the normal order In quantum field t ... [...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] |
Probabilistic Number Theory
In mathematics, Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underlying it is that different prime numbers are, in some serious sense, like independent random variables. This however is not an idea that has a unique useful formal expression. The founders of the theory were Paul Erdős, Aurel Wintner and Mark Kac during the 1930s, one of the periods of investigation in analytic number theory. Foundational results include the Erdős–Wintner theorem and the Erdős–Kac theorem on additive functions. See also *Number theory *Analytic number theory *Areas of mathematics *List of number theory topics *List of probability topics *Probabilistic method *Probable prime In number theory, a probable prime (PRP) is an integer that satisfies a specific condition that is satisfied by all prime numbers, but which is not satisfied by most composite numbers. Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Order Of An Arithmetic Function
In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let ''f'' be a function on the natural numbers. We say that ''g'' is a normal order of ''f'' if for every ''ε'' > 0, the inequalities : (1-\varepsilon) g(n) \le f(n) \le (1+\varepsilon) g(n) hold for ''almost all'' ''n'': that is, if the proportion of ''n'' ≤ ''x'' for which this does not hold tends to 0 as ''x'' tends to infinity. It is conventional to assume that the approximating function ''g'' is continuous and monotone. Examples * The Hardy–Ramanujan theorem: the normal order of ω(''n''), the number of distinct prime factors of ''n'', is log(log(''n'')); * The normal order of Ω(''n''), the number of prime factors of ''n'' counted with multiplicity, is log(log(''n'')); * The normal order of log(''d''(''n'')), where ''d''(''n'') is the number of divisors of ''n'', is log(2)&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed. Special case examples include the following: * All squares are rectangles (but not all rectangles are squares); therefore the square is a special case of the rectangle. * Fermat's Last Theorem, that has no solutions in positive integers with , is a special case of Beal's conjecture, that has no primitive solutions in positive integers with , , and all greater than 2, specifically, the case of {{mvar, x {{= y {{= z. See also * Specialization (logic) Specialization or Specializ ... [...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. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting 46 years and resulting in 28 joint papers. Life and education Turán was born into a Jewish family in Budapest on 18 August 1910.At the same period of time, Turán and Erdős were famous answerers in the journal '' KöMaL''. He received a teaching degree at the University of Budapest in 1933 and the PhD degree under Lipót Fejér in 1935 at Eötvös Loránd University. As a Jew, he fell victim to numerus clausus, and could not get a university job for several years. He was sent to labour service at various times from 1940-44. He is said to have been recognized and perhaps protected by a fascist guard, who, as a mathematics student, had admired Turán's work. Turán became associate professor at the University of Budapest in 1945 and ful ... [...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] |
Gérald Tenenbaum
Gérald Tenenbaum is a French mathematician and novelist, born in Nancy on 1 April 1952.Zéro faute à l’IUT Nancy-Brabois press release, University of Lorraine, January 30, 2012. Accessed on line June 26, 2012. He is one of the namesakes of the Erdős–Tenenbaum–Ford constant. Biography An alumnus of the , he has been professor of mathematics at the Institut Élie Cartan at ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Map
In algebra, an additive map, Z-linear map or additive function is a function f that preserves the addition operation: f(x + y) = f(x) + f(y) for every pair of elements x and y in the domain of f. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial. More formally, an additive map is a \Z-module homomorphism. Since an abelian group is a \Z-module, it may be defined as a group homomorphism between abelian groups. A map V \times W \to X that is additive in each of two arguments separately is called a bi-additive map or a \Z-bilinear map. Examples Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring. If f and g are additive maps, then the map f + g (defined pointwise) is additiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of ''n''". An example of an arithmetic function is the divisor function whose value at a positive integer ''n'' is equal to the number of divisors of ''n''. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions. This article provides links to functions of both classes. Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. Multiplicative and additive functions An arithmetic function ''a'' is * completely additive if ''a''(''mn'') = ''a''(''m'') + ''a''(''n'') for all natural numbers ''m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy–Ramanujan Theorem
In mathematics, the Hardy–Ramanujan theorem, proved by , states that the normal order of the number ω(''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 ''ψ''(''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 infinitesimal proportion of) integers. That is, let ''g''(''x'') be the number of positive integers ''n'' less than ''x'' for which the above inequality fails: then ''g''(''x'')/''x'' converges to zero as ''x'' goes to infinity. History A simple proof to the result was given by |
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] |
Joseph H
Joseph is a common male given name, derived from the Hebrew Yosef (יוֹסֵף). "Joseph" is used, along with "Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese and Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled '' Yūsuf''. In Persian, the name is "Yousef". The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with ''Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especially common in contemporary Israel, as either "Yossi" or "Yossef", and in Italy, where the name "Giuseppe" was the most common male name in the 20th century. In the first century CE, Joseph was the second most popular male name for Palestine Jews. In the Book of Genesis Joseph is Jacob's eleventh son and Rachel's first son, and k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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