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On The Number Of Primes Less Than A Given Magnitude
" die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin''. Overview This paper studies the prime-counting function using analytic methods. Although it is the only paper Riemann ever published on number theory, it contains ideas which influenced thousands of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. Among the new definitions, ideas, and notation introduced: *The use of the Greek letter zeta (ζ) for a function previously mentioned by Euler *The analytic continuation of this zeta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ueber Die Anzahl Der Primzahlen Unter Einer Gegebenen Grösse
" die Anzahl der Primzahlen unter einer gegebenen " (usual English translation: "On the Number of Primes Less Than a Given Magnitude") is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the ''Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin''. Overview This paper studies the prime-counting function using analytic methods. Although it is the only paper Riemann ever published on number theory, it contains ideas which influenced thousands of researchers during the late 19th century and up to the present day. The paper consists primarily of definitions, heuristic arguments, sketches of proofs, and the application of powerful analytic methods; all of these have become essential concepts and tools of modern analytic number theory. Among the new definitions, ideas, and notation introduced: *The use of the Greek letter zeta (ζ) for a function previously mentioned by Euler *The analytic continuation of this zeta func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler
Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential discoveries in many other branches of mathematics, such as analytic number theory, complex analysis, and infinitesimal calculus. He also introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is known for his work in mechanics, fluid dynamics, optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. Euler is credited for popularizing the Greek letter \pi (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Inversion
Fourier may refer to: * Fourier (surname), French surname Mathematics *Fourier series, a weighted sum of sinusoids having a common period, the result of Fourier analysis of a periodic function *Fourier analysis, the description of functions as sums of sinusoids *Fourier transform, the type of linear canonical transform that is the generalization of the Fourier series *Fourier operator, the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform *Fourier inversion theorem, any one of several theorems by which Fourier inversion recovers a function from its Fourier transform *Short-time Fourier transform or short-term Fourier transform (STFT), a Fourier transform during a short term of time, used in the area of signal analysis *Fractional Fourier transform (FRFT), a linear transformation generalizing the Fourier transform, used in the area of harmonic analysis *Discrete-time Fourier transform (DTFT), the reverse of the Fourier series, a special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Methods Of Contour Integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods. It also has various applications in physics. Contour integration methods include: * direct integration of a complex-valued function along a curve in the complex plane * application of the Cauchy integral formula * application of the residue theorem One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums. Curves in the complex plane In complex analysis, a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of singularities. The case of several complex variables is rather different, since singularities then need not be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Suppose ''f'' is an analytic function defined on a non-empty open subset ''U'' of the complex plane If ''V'' is a larger open subset of containing ''U'', and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ''G''(A''F''), for an algebraic group ''G'' and an algebraic number field ''F'', is a complex-valued function on ''G''(A''F'') that is l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ''ζ''(''s'') is a function whose argument ''s'' may be any complex numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Equation (number Theory)
Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional symptom ** Functional disorder * Functional classification for roads * Functional organization * Functional training In mathematics * Functional (mathematics), a term applied to certain scalar-valued functions in mathematics and computer science ** Minnesota functionals ** Functional analysis, a branch of mathematical analysis ** Linear functional, a type of functional often simply called a functional in the context of functional analysis * Higher-order function, also called a functional, a function that takes other functions as arguments In computer science, software engineering * "Functional" (noun) may be used as a synonym for Higher-order function * (C++), a header file in the C++ Standard Library * Functional design, a paradi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gamma Function
In mathematics, the gamma function (represented by Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined for all complex numbers z except non-positive integers, and for every positive integer z=n, \Gamma(n) = (n-1)!\,.The gamma function can be defined via a convergent improper integral for complex numbers with positive real part: \Gamma(z) = \int_0^\infty t^ e^\textt, \ \qquad \Re(z) > 0\,.The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic function, holomorphic except at zero and the negative integers, where it has simple Zeros and poles, poles. The gamma function has no zeros, so the reciprocal gamma function is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entire Function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a root at w, then f(z)/(z-w), taking the limit value at w, is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Just as meromorphic functions can be viewed as a generalization of rational fractions, entire functions can be viewed as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
