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 ...

, a zeta function is (usually) a function 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 \operatorname(s) > ...

\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 *

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\m ...

* Arithmetic zeta function *

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 th ...

of a dynamical system * Barnes zeta function 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 introd ...

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 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. : \ma ...

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

Lefschetz zeta function In mathematics, the Lefschetz zeta function, zeta-function is a tool used in topological periodic and fixed point (mathematics), fixed point theory, and dynamical systems. Given a continuous map f\colon X\to X, the zeta-function is defined as the fo ...

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 publis ...

, 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 is a ...

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 ...

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, and ...

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 *

, 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 fix ...

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 * 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 also a ...

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