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In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function

ζ ζ ( s ) = ∑ ∑ n = 1 ∞ ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.}

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 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 of a Riemann surface

• Shimizu L-function

• Shintani zeta function

• Subgroup zeta function

• Witten zeta function 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

• See also

• External links

• Jacobi zeta function

• Weierstrass zeta function

• Artin conjecture

• Birch and Swinnerton-Dyer conjecture

• Riemann hypothesis and the generalized Riemann hypothesis.

• Selberg class S

• Explicit formulae for L-functions

• Trace formula

• A directory of all known zeta functions