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Explicit Formula (L-function)
In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field. Riemann's explicit formula In his 1859 paper "On the Number of Primes Less Than a Given Magnitude" Riemann sketched an explicit formula (it was not fully proven until 1895 by von Mangoldt, see below) for the normalized prime-counting function which is related to the prime-counting function by :\pi_0(x) = \frac \lim_ \left ,\pi(x+h) + \pi(x-h)\,\right,, which takes the arithmetic mean of the limit from the left and the limit from the right at discontinuities. His formula was given in terms of the related function :f(x) = \pi_0(x) + \frac\,\pi_0(x^) + \frac\,\pi_0(x^) + \cdots in which a prime power counts as of a prime. ... [...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] |
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] |
Dirichlet L-function
In mathematics, a Dirichlet L-series is a function of the form :L(s,\chi) = \sum_^\infty \frac. where \chi is a Dirichlet character and s a complex variable with real part greater than 1 . It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L -function and also denoted L ( s , \chi) . These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the Dirichlet's theorem on arithmetic progressions, theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L ( s , \chi) is non-zero at s = 1 . Moreover, if \chi is principal, then the corresponding Dirichlet L -function has a simple pole at s = 1 . Otherwise, the L -function is entire function, entire. Euler product Since a Dirichlet character \chi is completely multiplicative, its L -function can also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Mangoldt Function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicative nor additive. Definition The von Mangoldt function, denoted by , is defined as :\Lambda(n) = \begin \log p & \textn=p^k \text p \text k \ge 1, \\ 0 & \text \end The values of for the first nine positive integers (i.e. natural numbers) are :0 , \log 2 , \log 3 , \log 2 , \log 5 , 0 , \log 7 , \log 2 , \log 3, which is related to . Properties The von Mangoldt function satisfies the identityApostol (1976) p.32Tenenbaum (1995) p.30 :\log(n) = \sum_ \Lambda(d). The sum is taken over all integers that divide . This is proved by the fundamental theorem of arithmetic, since the terms that are not powers of primes are equal to . For example, consider the case . Then :\begin \sum_ \Lambda(d) &= \Lambda(1) + \Lambda(2) + \Lambda(3) + \Lambda(4) + \Lambda(6) + \Lambda(12) \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function. Definition In general, if is a bounded multiplicative function, then the Dirichlet series :\sum_^\infty \frac is equal to :\prod_ P(p, s) \quad \text \operatorname(s) >1 . where the product is taken over prime numbers , and is the sum :\sum_^\infty \frac = 1 + \frac + \frac + \frac + \cdots In fact, if we consider these as formal generating functions, the existence of such a ''formal'' Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes . An important special c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function is defined by the formula \frac where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the infinitesimal absolute change in , namely scaled by the current value of . When is a function of a real variable , and takes real, strictly positive values, this is equal to the derivative of , or the natural logarithm of . This follows directly from the chain rule: \frac\ln f(x) = \frac \frac Basic properties Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitary matrices. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the concept of isomorphism ''between'' Hilbert spaces. Definition Definition 1. A ''unitary operator'' is a bounded linear operator on a Hilbert space that satisfies , where is the adjoint of , and is the identity operator. The weaker condition defines an ''isometry''. The other weaker condition, , defines a ''coisometry''. Thus a unitary operator is a bounded linear operator that is both an isometry and a coisometry, or, equivalently, a surjective isometry. An equivalent definition is the following: Definition 2. A ''unitary operator'' is a bounded linear operator on a Hilbert space for which the followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digamma Function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: :\psi(z) = \frac\ln\Gamma(z) = \frac. It is the first of the polygamma functions. This function is Monotonic function, strictly increasing and Concave function, strictly concave on (0,\infty), and it Asymptotic analysis, asymptotically behaves as :\psi(z) \sim \ln - \frac, for complex numbers with large modulus (, z, \rightarrow\infty) in the Circular sector, sector , \arg z, 0. The digamma function is often denoted as \psi_0(x), \psi^(x) or (the uppercase form of the archaic Greek consonant digamma meaning Gamma, double-gamma). Gamma. Relation to harmonic numbers The gamma function obeys the equation :\Gamma(z+1)=z\Gamma(z). \, Taking the logarithm on both sides and using the functional equation property of the log-gamma function gives: :\log \Gamma(z+1)=\log(z)+\log \Gamma(z), Differentiating both sides with respect to gives: :\psi(z+1)=\psi(z)+\frac Since the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was one of the most influential mathematicians of the twentieth century. His influence is due both to his original contributions to a remarkably broad spectrum of mathematical theories, and to the mark he left on mathematical practice and style, through some of his own works as well as through the Bourbaki group, of which he was one of the principal founders. Life André Weil was born in Paris to agnostic Alsatian Jewish parents who fled the annexation of Alsace-Lorraine by the German Empire after the Franco-Prussian War in 1870–71. Simone Weil, who would later become a famous philosopher, was Weil's younger sister and only sibling. He studied in Paris, Rome and Göttingen and received his doctorate in 1928. While in Germany, Weil befriended Carl Ludwig Siegel. Starting in 1930, he spent two academic years at Aligarh Mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if the base is implicit, simply . Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity. The natural logarithm of is the exponentiation, power to which would have to be raised to equal . For example, is , because . The natural logarithm of itself, , is , because , while the natural logarithm of is , since . The natural logarithm can be defined for any positive real number as the Integral, area under the curve from to (with the area being negative when ). The simplicity of this definition, which is matched in many other formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
