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Landsberg–Schaar Relation
In number theory and harmonic analysis, the Landsberg–Schaar relation (or identity) is the following equation, which is valid for arbitrary positive integers ''p'' and ''q'': : \frac\sum_^\exp\left(\frac\right)= \frac\sum_^\exp\left(-\frac\right). The standard way to prove it is to put = + ''ε'', where ''ε'' > 0 in this identity due to Jacobi (which is essentially just a special case of the Poisson summation formula in classical harmonic analysis): : \sum_^e^=\frac \sum_^e^ and then let ''ε'' → 0. A proof using only finite methods was discovered in 2018 by Ben Moore. If we let ''q'' = 1, the identity reduces to a formula for the quadratic Gauss sum In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; f ... modulo ''p''. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency. The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals. Generalizing these transforms to other domains is generally called Fourier analysis, although the term is sometimes used interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis, spectral analysis, and neuroscience. The term "harmonics" originated from the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the freq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants and number theory. Biography Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of four children of a banker, Simon Jacobi. His elder brother, Moritz, would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were taught all the standard subjects: classical languages, history, philology, mathematics, sciences, etc. As a result of the good education he had received from his uncle, as well as his own remarkable abilities, after less than half a year Jacobi was moved to the senior year despite his young age. However, as the Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Summation Formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function (mathematics), function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation. For a smooth, complex valued function s(x) on \mathbb R which decays at infinity with all derivatives (Schwartz function), the simplest version of the Poisson summation formula states that where S is the Fourier transform of s, i.e., S(f) \triangleq \int_^ s(x)\ e^\, dx. The summation formula can be restated in many equivalent ways, but a simple one is the following. Sup ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Gauss Sum
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws. Definition For an odd prime number and an integer , the quadratic Gauss sum is defined as : g(a;p) = \sum_^\zeta_p^, where \zeta_p is a primitive th root of unity, for example \zeta_p=\exp(2\pi i/p). Equivalently, : g(a;p) = \sum_^\big(1+\left(\tfrac\right)\big)\,\zeta_p^. For divisible by , and we have \zeta_p^=1 and thus : g(a;p) = p. For not divisible by , we have \sum_^ \zeta_p^ = 0, implying that : g(a;p) = \sum_^\left(\tfrac\right)\,\zeta_p^ = G(a,\left(\tfrac\right)), where : G(a,\chi)=\sum_^\chi(n)\,\zeta_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
