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 ar ...

, a

zeta Zeta (, ; uppercase Ζ, lowercase ζ; , , 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 from the Phoenician alphabet, Phoenician letter zay ...

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 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 and its analytic c ...

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

Zeta functions include: * Airy zeta function, 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 Ai(''x'') and the related function Bi(''x''), are Linear in ...

* Arakawa–Kaneko zeta function *

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 arith ...

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 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 ...

or double zeta function * Beurling zeta function 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 ca ...

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 ma ...

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 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 c ...

, a generalization of the Riemann zeta function * Igusa zeta function *

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 first ...

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

Lerch zeta function In mathematics, the Lerch transcendent, 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 a similar function in 1887. The Ler ...

, a generalization of the Riemann zeta function *

Local zeta function In mathematics, 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^)^k\right) where is a Singular point of an algebraic variety, non-s ...

of a characteristic-''p'' variety * Matsumoto zeta function *

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 of a motive * Multiple zeta function, 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 *

, the archetypal example * Ruelle zeta function *

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 In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

* Shimizu ''L''-function * Shintani zeta function * 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 (mathematics ...

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