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List Of Zeta Functions
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * Arakawa–Kaneko zeta function * Arithmetic zeta function * Artin–Mazur zeta function of a dynamical system * Barnes zeta function or double zeta function * Beurling zeta function of Beurling generalized primes * Dedekind zeta function of a number field * Duursma zeta function of error-correcting codes * Epstein zeta function of a quadratic form * Goss zeta function of a function field * Hasse–Weil zeta function of a variety * Height zeta function of a variety * Hurwitz zeta function, a generalization of the Riemann zeta function * Igusa zeta function * Ihara zeta function of a graph * ''L''-function, a "twisted" zeta function * Lefschetz zeta function of a morphism * Lerch zeta function, a generalization of the Riemann zeta functio ... [...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] |
Igusa Zeta Function
In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, ''modulo'' ''p'', ''p''2, ''p''3, and so on. Definition For a prime number ''p'' let ''K'' be a p-adic field, i.e. : \mathbb_p\infty , ''R'' the valuation ring and ''P'' the maximal ideal. For z \in K we denote by \operatorname(z) the valuation of ''z'', \mid z \mid = q^, and ac(z)=z \pi^ for a uniformizing parameter π of ''R''. Furthermore let \phi : K^n \to \mathbb be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let \chi be a character of R^\times. In this situation one associates to a non-constant polynomial f(x_1, \ldots, x_n) \in K _1,\ldots,x_n/math> the Igusa zeta function : Z_\phi(s,\chi) = \int_ \phi(x_1,\ldots,x_n) \chi(ac(f(x_1,\ldots,x_n))) , f(x_1,\ldots,x_n), ^s \, dx where s \in \mathbb, \operatorname(s)>0, and ''dx'' is Haar measure so normalized that R^n has measure 1. Igusa's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg Zeta Function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the primes numbers. If \Gamma is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows, :\zeta_\Gamma(s)=\prod_p(1-N(p)^)^, or :Z_\Gamma(s)=\prod_p\prod^\infty_(1-N(p)^), where ''p'' runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of \Gamma), and ''N''(''p'') denotes the length of ''p'' (equivalently, the square of the bigger eigenvalue of ''p''). For any hyperbolic surface of finite area there is an associated Selberg zeta-function; this function is a meromorphic function defined in the complex plane. The zeta function is defined in terms of the closed geodesics of the surface. The zeros and poles of the Selberg zeta-function, ''Z''(''s' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruelle Zeta Function
In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle. Formal definition Let ''f'' be a function defined on a manifold ''M'', such that the set of fixed points Fix(''f'' ''n'') is finite for all ''n'' > 1. Further let ''φ'' be a function on ''M'' with values in ''d'' × ''d'' complex matrices. The zeta function of the first kind isTerras (2010) p. 28 : \zeta(z) = \exp\left( \sum_ \frac \sum_ \operatorname \left( \prod_^ \varphi(f^k(x)) \right) \right) Examples In the special case ''d'' = 1, ''φ'' = 1, we have : \zeta(z) = \exp\left( \sum_ \frac m \left, \operatorname(f^m)\ \right) which is the Artin–Mazur zeta function. The Ihara zeta function In mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Zeta Function
In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for \Re(s) > 1: :P(s)=\sum_ \frac=\frac+\frac+\frac+\frac+\frac+\cdots. Properties The Euler product for the Riemann zeta function ''ζ''(''s'') implies that : \log\zeta(s)=\sum_ \frac n which by Möbius inversion gives :P(s)=\sum_ \mu(n)\frac n When ''s'' goes to 1, we have P(s)\sim \log\zeta(s)\sim\log\left(\frac \right). This is used in the definition of Dirichlet density. This gives the continuation of ''P''(''s'') to \Re(s) > 0, with an infinite number of logarithmic singularities at points ''s'' where ''ns'' is a pole (only ''ns'' = 1 when ''n'' is a squarefree number greater than or equal to 1), or zero of the Riemann zeta function ''ζ''(.). The line \Re(s) = 0 is a natural boundary as the singularities cluster near all points of this line. If one defines a sequence :a_n=\prod_ \frac=\prod_ \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic L-function
In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are ''p-adic'' (where ''p'' is a prime number). For example, the domain could be the ''p''-adic integers Z''p'', a profinite ''p''-group, or a ''p''-adic family of Galois representations, and the image could be the ''p''-adic numbers Q''p'' or its algebraic closure. The source of a ''p''-adic ''L''-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a ''p''-adic ''L''-function —is via the ''p''-adic interpolation of special values of ''L''-functions. For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a ''p''-adic ''L''-function, the ''p''-adic Riemann zeta function ζ''p''(''s''), whose values at negative odd integers are those of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Zeta Function
In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by :\zeta(s_1,\ldots,s_k) = \sum_\ \frac = \sum_\ \prod_^k \frac,\! and converge when Re(''s''1) + ... + Re(''s''''i'') > ''i'' for all ''i''. Like the Riemann zeta function, the multiple zeta functions can be analytically continued to be meromorphic functions (see, for example, Zhao (1999)). When ''s''1, ..., ''s''''k'' are all positive integers (with ''s''1 > 1) these sums are often called multiple zeta values (MZVs) or Euler sums. These values can also be regarded as special values of the multiple polylogarithms. The ''k'' in the above definition is named the "depth" of a MZV, and the ''n'' = ''s''1 + ... + ''s''''k'' is known as the "weight". The standard shorthand for writing multiple zeta functions is to place repeating strings of the argument within braces and use a superscript to indicate the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivic Zeta Function
In algebraic geometry, the motivic zeta function of a smooth algebraic variety X is the formal power series :Z(X,t)=\sum_^\infty ^^n Here X^ is the n-th symmetric power of X, i.e., the quotient of X^n by the action of the symmetric group S_n, and ^/math> is the class of X^ in the ring of motives (see below). If the ground field is finite, and one applies the counting measure to Z(X,t), one obtains the local zeta function of X. If the ground field is the complex numbers, and one applies Euler characteristic with compact supports to Z(X,t), one obtains 1/(1-t)^. Motivic measures A motivic measure is a map \mu from the set of finite type schemes over a field k to a commutative ring A, satisfying the three properties :\mu(X)\, depends only on the isomorphism class of X, :\mu(X)=\mu(Z)+\mu(X\setminus Z) if Z is a closed subscheme of X, :\mu(X_1\times X_2)=\mu(X_1)\mu(X_2). For example if k is a finite field and A= is the ring of integers, then \mu(X)=\#(X(k)) defines a motivic m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minakshisundaram–Pleijel Zeta Function
The Minakshisundaram–Pleijel zeta function is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold. It was introduced by . The case of a compact region of the plane was treated earlier by . Definition For a compact Riemannian manifold ''M'' of dimension ''N'' with eigenvalues \lambda_1, \lambda_2, \ldots of the Laplace–Beltrami operator \Delta, the zeta function is given for \operatorname(s) sufficiently large by : Z(s) = \mbox(\Delta^) = \sum_^ \vert \lambda_ \vert^. (where if an eigenvalue is zero it is omitted in the sum). The manifold may have a boundary, in which case one has to prescribe suitable boundary conditions, such as Dirichlet or Neumann boundary conditions. More generally one can define : Z(P, Q, s) = \sum_^ \frac for ''P'' and ''Q'' on the manifold, where the f_n are normalized eigenfunctions. This can be analytically continued to a meromorphic function of ''s'' for all complex ''s'', and is holomorphic for P\ne Q. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matsumoto Zeta Function
In mathematics, Matsumoto zeta functions are a type of zeta function introduced by Kohji Matsumoto in 1990. They are functions of the form :\phi(s)=\prod_\frac where ''p'' is a prime and ''A''''p'' is a polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa .... References * Zeta and L-functions {{numtheory-stub ... [...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] |
Lerch Zeta Function
In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887. Definition The Lerch zeta function is given by :L(\lambda, s, \alpha) = \sum_^\infty \frac . A related function, the Lerch transcendent, is given by :\Phi(z, s, \alpha) = \sum_^\infty \frac . The two are related, as :\,\Phi(e^, s,\alpha)=L(\lambda, s, \alpha). Integral representations The Lerch transcendent has an integral representation: : \Phi(z,s,a)=\frac\int_0^\infty \frac\,dt The proof is based on using the integral definition of the Gamma function to write :\Phi(z,s,a)\Gamma(s) = \sum_^\infty \frac \int_0^\infty x^s e^ \frac = \sum_^\infty \int_0^\infty t^s z^n e^ \frac and then interchanging the sum and integral. The resulting integral representation converges for z \in \Complex \setm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
