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Dirichlet Divisor Problem
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems. Definition The divisor summatory function is defined as :D(x)=\sum_ d(n) = \sum_ 1 where :d(n)=\sigma_0(n) = \sum_ 1 is the divisor function. The divisor function counts the number of ways that the integer ''n'' can be written as a product of two integers. More generally, one defines :D_k(x)=\sum_ d_k(n)= \sum_\sum_ d_(n) where ''d''''k''(''n'') counts the number of ways that ''n'' can be written as a product of ''k'' numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in ''k'' dimensions. Thus, for ''k'' = 2, ''D''(''x'') = ''D''2(''x'') counts the number of points on a square lattice bounded on the left by the v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigori Kolesnik (mathematician)
Grigory, Grigori and Grigoriy () are Russian masculine given names. Russian version of Gregory (given name). Grigory * Grigory Baklanov (1923–2009), Russian novelist * Grigory Barenblatt (1927–2018), Russian mathematician * Grigory Bey-Bienko (1903–1971), Russian entomologist * Grigory Danilevsky (1829–1890), Russian novelist * Grigory Falko (born 1987), Russian swimmer * Grigory Fedotov (1916–1957), Soviet football player and manager * Grigory Frid (1915–2012), Russian composer * Grigory Gagarin (1810–1893), Russian painter and military commander * Grigory Gamarnik (1929–2018), Soviet wrestler * Grigory Gamburtsev (1903–1955), Soviet seismologist * Grigory Ginzburg (1904–1961), Russian pianist * Grigory Grum-Grshimailo (1860–1936), Russian entomologist * Grigory Gurkin (1870–1937), Altay landscape painter * Grigory Helbach (1863–1930), Russian chess master * Grigory Kaminsky (1894–1938), Soviet politician * Grigory Kiriyenko (born 1965), Russian fe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Functions
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 the requirement that an arithmetical function "expresses some arithmetical property of ''n''". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes. 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''. 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'' and ''n''; * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Edwards (mathematician)
Harold Mortimer Edwards, Jr. (August 6, 1936 – November 10, 2020) was an American mathematician working in number theory, algebra, and the history and philosophy of mathematics. He was one of the co-founding editors, with Bruce Chandler, of ''The Mathematical Intelligencer''. He is the author of expository books on the Riemann zeta function, on Galois theory, and on Fermat's Last Theorem. He wrote a book on Leopold Kronecker's work on divisor theory providing a systematic exposition of that work—a task that Kronecker never completed. He wrote textbooks on linear algebra, calculus, and number theory. He also wrote a book of essays on constructive mathematics. Edwards graduated from the University of Wisconsin–Madison in 1956, received a Master of Arts from Columbia University in 1957, and a Ph.D from Harvard University in 1961, under the supervision of Raoul Bott. He taught at Harvard and Columbia University; he joined the faculty at New York University in 1966, and was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Integral Formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis. Theorem Let be an open subset of the complex plane , and suppose the closed disk defined as D = \bigl\ is completely contained in . Let be a holomorphic function, and let be the circle, oriented counterclockwise, forming the boundary of . Then for every in the interior of , f(a) = \frac \oint_\gamma \frac\,dz.\, The proof of this statement uses the Cauchy integral theorem and like that theorem, it only requires to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue (complex Analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function f\colon \mathbb \setminus \_k \rightarrow \mathbb that is holomorphic except at the discrete points ''k'', even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem. Definition The residue of a meromorphic function f at an isolated singularity a, often denoted \operatorname(f,a), \operatorname_a(f), \mathop_f(z) or \mathop_f(z), is the unique value R such that f(z)- R/(z-a) has an analytic antiderivative in a punctured disk 0<\vert z-a\vert<\delta. Alternatively, residues can be calculated by finding |
Mellin Transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a complex-valued function defined on \mathbf R^_+= (0,\infty) is the function \mathcal M f of complex variable s given (where it exists, see Fundamental strip below) by \mathcal\left\(s) = \varphi(s)=\int_0^\infty x^ f(x) \, dx = \int_f(x) x^s \frac. Notice that dx/x is a Haar measure on the multiplicative group \mathbf R^_+ and x\mapsto x^s is a (in general non-unitary) multiplicative character. The inverse transform is \mathcal^\left\(x) = f(x)=\frac \int_^ x^ \varphi(s)\, ds. The notation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandar Ivić
Aleksandar Ivić (March 6, 1949 – December 27, 2020) was a Serbian mathematician, specializing in analytic number theory. He gained an international reputation and gave lectures on the Riemann zeta function at universities around the world. (in English) Biography Aleksandar Ivić was born in Belgrade to two renowned linguists, the academician Pavle Ivić (1924–1999) and the academician Milka Ivić (1923–2011). His paternal grandfather was the historian Aleksa Ivić (1881–1948) and his maternal great-grandfather was the poet Vojislav Ilić (1862–1894), the son of the writer and minister of justice Jovan Ilić (1824–1901). In 1967, Aleksandar Ivić successfully participated in the International Mathematical Olympiad. He graduated in 1971 from the University of Novi Sad with a bachelor's degree in science. As a graduate student in Faculty of Sciences of the University of Belgrade, he graduated with a master's degree in 1973 and a doctorate in 1975. His doctoral dissertati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adolf Piltz
Adolf Piltz (8 December 1855 – 1940) was a German mathematician who contributed to number theory. Piltz was arguably the first to formulate a generalized Riemann hypothesis, in 1884.Davenport, p. 124. Notes References *Harold Davenport, Davenport, Harold. ''Multiplicative number theory''. Third edition. Revised and with a preface by Hugh L. Montgomery. Graduate Texts in Mathematics, 74. Springer-Verlag, New York, 2000. xiv+177 pp. . Further reading * External links * 1855 births 1940 deaths 19th-century German mathematicians Humboldt University of Berlin alumni 20th-century German mathematicians Mathematicians from the German Empire {{Germany-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree Of A Polynomial
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the highest exponent occurring in the polynomial. The term order has been used as a synonym of ''degree'' but, nowadays, may refer to several other concepts (see Order of a polynomial (other)). For example, the polynomial 7x^2y^3 + 4x - 9, which can also be written as 7x^2y^3 + 4x^1y^0 - 9x^0y^0, has three terms. The first term has a degree of 5 (the sum of the powers 2 and 3), the second term has a degree of 1, and the last term has a degree of 0. Therefore, the polynomial has a degree of 5, which is the highest degree of any term. To determine the degree of a polynomial that is not in standard form, such as (x+1)^2 - (x-1)^2, one c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Exponent Pairs
In mathematics, van der Corput's method generates estimates for exponential sums. The method applies two processes, the van der Corput processes A and B which relate the sums into simpler sums which are easier to estimate. The processes apply to exponential sums of the form : \sum_^b e(f(n)) \ where ''f'' is a sufficiently smooth function and ''e''(''x'') denotes exp(2πi''x''). Process A To apply process A, write the first difference ''f''''h''(''x'') for ''f''(''x''+''h'')−''f''(''x''). Assume there is ''H'' ≤ ''b''−''a'' such that : \sum_^H \left\vert\right\vert \le b-a \ . Then : \left\vert\right\vert \ll \frac \ . Process B Process B transforms the sum involving ''f'' into one involving a function ''g'' defined in terms of the derivative of f. Suppose that ''f is monotone increasing with ''f'''(''a'') = α, ''f'''(''b'') = β. Then ''f''' is invertible on �,βwith inverse ''u'' say. Further suppose ''f'''' ≥ λ > 0. Write : g(y) = f(u(y)) - y u(y) \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
