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Hardy–Ramanujan–Littlewood Circle Method
In mathematics, the Hardy–Ramanujan–Littlewood circle method is a technique of analytic number theory. It is named for G. H. Hardy, S. Ramanujan, and J. E. Littlewood, who developed it in a series of papers on Waring's problem. History The initial idea is usually attributed to the work of Hardy with Srinivasa Ramanujan a few years earlier, in 1916 and 1917, on the asymptotics of the partition function. It was taken up by many other researchers, including Harold Davenport and I. M. Vinogradov, who modified the formulation slightly (moving from complex analysis to exponential sums), without changing the broad lines. Hundreds of papers followed, and the method still yields results. The method is the subject of a monograph by R. C. Vaughan. Outline The goal is to prove asymptotic behavior of a series: to show that for some function. This is done by taking the generating function of the series, then computing the residues about zero (essentially the Fourier coefficients). ... [...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] |
Unit Circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere. If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation x^2 + y^2 = 1. Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant. The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk. One may also use other notions of "dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube Roots Of Unity
In mathematics, a root of unity is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform. It is occasionally called a de Moivre number after French mathematician Abraham de Moivre. Roots of unity can be defined in any field. If the characteristic of the field is zero, the roots are complex numbers that are also algebraic integers. For fields with a positive characteristic, the roots belong to a finite field, and, conversely, every nonzero element of a finite field is a root of unity. Any algebraically closed field contains exactly th roots of unity, except when is a multiple of the (positive) characteristic of the field. General definition An ''th root of unity'', where is a positive integer, is a number satisfying the equation z^n = 1. Unless otherwise specified, the roots of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
