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Selberg Class
In mathematics, the Selberg class is an axiomatic definition of a class of ''L''-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called ''L''-functions or zeta functions. Although the exact nature of the class is conjectural, the hope is that the definition of the class will lead to a classification of its contents and an elucidation of its properties, including insight into their relationship to automorphic forms and the Riemann hypothesis. The class was defined by Atle Selberg in , who preferred not to use the word "axiom" that later authors have employed. Definition The formal definition of the class ''S'' is the set of all Dirichlet series :F(s)=\sum_^\infty \frac absolutely convergent for Re(''s'') > 1 that satisfy four axioms (or assumptions as Selberg calls them): Comments on definition The condition that the real part of μ''i'' ... [...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] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > 1 and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory, and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Conjectures
In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as "a kind of grand unified theory of mathematics." The Langlands program consists of some very complicated theoretical abstractions, which can be difficult even for specialist mathematicians to grasp. To oversimplify, the fundamental lemma of the project posits a direct connection between the generalized fundamental representation of a finite field with its group extension to the automorphic forms under which it is invariant. This is accomplished through abstraction to higher dimensional integration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvable Extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Class field theory provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either definition, is always abelian. If a field ''K'' contains a primitive ''n''-th root of unity and the ''n''-th ro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. Definition Suppose that E is an extension of the field F (written as E/F and read "''E'' over ''F'' "). An automorphism of E/F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E/F is an isomorphism \alpha:E\to E such that \alpha(x) = x for each x\in F. The set of all automorphisms of E/F forms a group with the operation of function composition. This group is sometimes denoted by \operatorname(E/F). If E/F is a Galois extension, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin L-function
In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin ''L''-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis. Definition Given \rho , a representation of G on a finite-dimensional complex vector space V, where G is the Galois group of the finite extension L/K of number fields, the Artin L-function: L(\rho,s) is defined by an Euler product. For each prime ideal \mathfrak p in K's ring of integers, there is an Euler factor, whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin Conjecture (L-functions)
In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin ''L''-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis. Definition Given \rho , a representation of G on a finite-dimensional complex vector space V, where G is the Galois group of the finite extension L/K of number fields, the Artin L-function: L(\rho,s) is defined by an Euler product. For each prime ideal \mathfrak p in K's ring of integers, there is an Euler factor, whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Representation
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 le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuspidal Representation
In number theory, cuspidal representations are certain representations of algebraic groups that occur discretely in L^2 spaces. The term ''cuspidal'' is derived, at a certain distance, from the cusp forms of classical modular form theory. In the contemporary formulation of automorphic representations, representations take the place of holomorphic functions; these representations may be of adelic algebraic groups. When the group is the general linear group \operatorname_2, the cuspidal representations are directly related to cusp forms and Maass forms. For the case of cusp forms, each Hecke eigenform ( newform) corresponds to a cuspidal representation. Formulation Let ''G'' be a reductive algebraic group over a number field ''K'' and let A denote the adeles of ''K''. The group ''G''(''K'') embeds diagonally in the group ''G''(A) by sending ''g'' in ''G''(''K'') to the tuple (''g''''p'')''p'' in ''G''(A) with ''g'' = ''g''''p'' for all (finite and infinite) primes ''p''. Let ''Z'' d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W), with W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field K of arbitrary characteristic, rather than a vector space over the field of real numbers or o ... [...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'', ''χ''). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that is non-zero at ''s'' = 1. Moreover, if ''χ'' is principal, then the corresponding Dirichlet ''L''-function has a simple pole at ''s'' = 1. Otherwise, the ''L''-function is entire. Euler product Since a Dirichlet character ''χ'' is completely multiplicative, its ''L''-function can also be written as an Euler product in the half-plane of absol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
