In mathematics, a

zeta Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived fr ...

function is (usually) a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

analogous to the original example, the Riemann zeta function :

\zeta(s) = \sum_^\infty \frac 1 .

Zeta functions include: *

Airy zeta function In mathematics, the Airy zeta function, studied by , is a function analogous to the Riemann zeta function and related to the zeros of the Airy function. Definition The Airy function :\mathrm(x) = \frac \int_0^\infty \cos\left(\tfrac13t^3 + xt\rig ...

, related to the zeros of the

Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent soluti ...

Arakawa–Kaneko zeta function In mathematics, the Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function. Definition The zeta function \xi_k(s) is defined by :\xi_k(s) = \frac \int_0^ \frac ...

Arithmetic zeta function In mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes the Riemann zeta function and Dedekind zeta function to higher dimensions. The arithme ...

Artin–Mazur zeta function In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. It is defined from a given function f as t ...

of a dynamical system *

Barnes zeta function In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by . It is further generalized by the Shintani zeta function. Definition The Barnes zeta function is defined by : \zeta_N(s,w\mid a_1,\ldots,a_N)= ...

or double zeta function *

Beurling zeta function In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were intro ...

of Beurling generalized primes *

Dedekind zeta function In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ...

of a number field * Duursma zeta function of error-correcting codes *

Epstein zeta function In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are ...

of a quadratic form *

Goss zeta function In the field of mathematics, the Goss zeta function, named after David Goss, is an analogue of the Riemann zeta function for function fields. proved that it satisfies an analogue of the Riemann hypothesis. proved results for a higher-dimensiona ...

of a function field *

Hasse–Weil zeta function In mathematics, the Hasse–Weil zeta function attached to an algebraic variety ''V'' defined over an algebraic number field ''K'' is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reduci ...

of a variety *

Height zeta function In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of given height. Definition If ''S'' is a set with height function ''H'', such that there are only finitely ...

of a variety *

Hurwitz zeta function In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by :\zeta(s,a) = \sum_^\infty \frac. This series is absolutely convergent for the given values of and and ...

, a generalization of the Riemann zeta function *

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. : \m ...

Ihara zeta function In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was firs ...

of a graph * ''L''-function, a "twisted" zeta function *

Lefschetz zeta function In mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a continuous map f\colon X\to X, the zeta-function is defined as the formal series :\zeta_f(t) = \exp \left( ...

of a morphism *

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 publ ...

, a generalization of the Riemann zeta function *

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 algebr ...

of a characteristic-''p'' variety *

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 ...

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 ...

of a Laplacian *

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, a ...

of a motive *

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'')&nbs ...

, or Mordell–Tornheim zeta function of several variables * ''p''-adic zeta function of a ''p''-adic number *

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 ...

, like the Riemann zeta function, but only summed over primes * Riemann zeta function, the archetypal example *

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 ...

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 inste ...

of a Riemann surface * Shimizu ''L''-function *

Shintani zeta function In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by . They include Hurwitz zeta functions and Barnes zeta functions. Definition Let P(\mathbf) be a polynom ...

* Subgroup zeta function *

Witten zeta function In mathematics, the Witten zeta function, is a function associated to a root system that encodes the degrees of the irreducible representations of the corresponding Lie group In mathematics, a Lie group (pronounced ) is a group that is al ...

of a Lie group * Zeta function of an incidence algebra, a function that maps every interval of a poset to the constant value 1. Despite not resembling a holomorphic function, the special case for the poset of integer divisibility is related as a formal Dirichlet series to the Riemann zeta function. * Zeta function of an operator or spectral zeta function

External links

A directory of all known zeta functions
{{DEFAULTSORT:Zeta functions * Mathematics-related lists

See also

External links