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In [[mathematics]], a '''[[zeta]] function''' is (usually) a [[function (mathematics)|function]] analogous to the original example: the [[Riemann zeta function]]
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: <math>\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}.</math>
 
Zeta functions include:
* [[Airy zeta function]], related to the zeros of the [[Airy function]]
* [[Arithmetic zeta function]]
* [[Artin–Mazur 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|Dedekind zeta-function]] of a number field
* [[Real analytic Eisenstein series#Epstein zeta function|Epstein zeta-function]] of a quadratic form.
* [[Goss zeta function]] of a function field
* [[Hasse–Weil zeta function|Hasse–Weil zeta-function]] of a variety
* [[Height zeta function]] of a variety
* [[Hurwitz zeta function|Hurwitz zeta-function]] A generalization of the Riemann zeta function
* [[Ihara zeta function|Ihara zeta-function]] of a graph
* [[Igusa zeta function|Igusa zeta-function]]
* [[L-function]], a 'twisted'  zeta-function.  
* [[Lefschetz zeta function|Lefschetz zeta-function]] of a morphism
* [[Lerch zeta function|Lerch zeta-function]] A generalization of the Riemann zeta function
* [[Local zeta-function]] of a characteristic ''p'' variety
* [[Matsumoto zeta function]]
* [[Minakshisundaram–Pleijel zeta function]] 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]] Like the Riemann zeta function, but only summed over primes.  
* [[Riemann zeta function]] The archetypal example.
* [[Ruelle zeta function]]
* [[Selberg zeta function|Selberg zeta-function]] of a Riemann surface
* [[Shimizu L-function]]
* [[Shintani zeta function]]
* Subgroup zeta function
* [[Witten zeta function]] of a Lie group
* [[Incidence algebra#Special elements|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 (operator)|Zeta function of an operator]] or Spectral zeta function
 
==See also==
;Other functions called zeta functions, but not analogous to the Riemann zeta function
*[[Jacobi zeta function]]
*[[Weierstrass zeta function]]
 
;Topics related to zeta functions
*[[Artin conjecture (L-functions)|Artin conjecture]]
*[[Birch and Swinnerton-Dyer conjecture]]
*[[Riemann hypothesis]] and the [[generalized Riemann hypothesis]].
*[[Selberg class]] S
 
==External links==
* [http://www.maths.ex.ac.uk/~mwatkins/zeta/directoryofzetafunctions.htm A directory of all known zeta functions]
 
[[Category:Zeta and L-functions|*]]
[[Category:Mathematics-related lists]]

Latest revision as of 02:21, 14 November 2014

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