|
Explicit Formulae (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] |
Chebyshev's Function
In mathematics, the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function or is given by :\vartheta(x)=\sum_ \ln p where \ln denotes the natural logarithm, with the sum extending over all prime numbers that are less than or equal to . The second Chebyshev function is defined similarly, with the sum extending over all prime powers not exceeding : \psi(x) = \sum_\sum_\ln p=\sum_ \Lambda(n) = \sum_\left\lfloor\log_p x\right\rfloor\ln p, where is the von Mangoldt function. The Chebyshev functions, especially the second one , are often used in proofs related to prime numbers, because it is typically simpler to work with them than with the prime-counting function, (See the exact formula, below.) Both Chebyshev functions are asymptotic to , a statement equivalent to the prime number theorem. Tchebycheff function, Chebyshev utility function, or weighted Tchebycheff scalarizing func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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 a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. The philosopher Simone Weil was his sister. The writer Sylvie Weil is his daughter. 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 Muslim University in India. Aside from mathematics, Weil held lifelong interests in classical Greek and Latin literature, in Hinduism and Sanskrit literature: he had taught himself Sanskrit in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Riemann Hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, which are formally similar to the Riemann zeta-function. One can then ask the same question about the zeros of these ''L''-functions, yielding various generalizations of the Riemann hypothesis. Many mathematicians believe these generalizations of the Riemann hypothesis to be true. The only cases of these conjectures which have been proven occur in the algebraic function field case (not the number field case). Global ''L''-functions can be associated to elliptic curves, number fields (in which case they are called Dedekind zeta-functions), Maass forms, and Dirichlet characters (in which case they are called Dirichlet L-functions). When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Zeta-function
In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right) where is a non-singular -dimensional projective algebraic variety over the field with elements and is the number of points of defined over the finite field extension of . Making the variable transformation gives : \mathit (V,u) = \exp \left( \sum_^ N_m \frac \right) as the formal power series in the variable u. Equivalently, the local zeta function is sometimes defined as follows: : (1)\ \ \mathit (V,0) = 1 \, : (2)\ \ \frac \log \mathit (V,u) = \sum_^ N_m u^\ . In other words, the local zeta function with coefficients in the finite field is defined as a function whose logarithmic derivative generates the number of solutions of the equation defining in the degree extension Formulation Given a finite field ''F'', there is, up to isomorphism, only one field ''Fk'' with : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert–Pólya Conjecture
In mathematics, the Hilbert–Pólya conjecture states that the non-trivial Zero of a function, zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis, by means of spectral theory. History In a letter to Andrew Odlyzko, dated January 3, 1982, George Pólya said that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts ''t'' of the zeros : \tfrac12 + it of the Riemann zeta function corresponded to eigenvalues of a self-adjoint operator.. The earliest published statement of the conjecture seems to be in .. David Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture due to a story told by Ernst Hellinger, a student of Hilbert, to André Weil. Hellinger said that Hilber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
