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

: <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
  • 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
  • Riemann–Siegel zeta function, or Hardy zeta function, alternative names for the Z function
  • 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

;Other functions called zeta functions, but not analogous to the Riemann zeta function

  • Jacobi zeta function
  • Weierstrass zeta function

;Topics related to zeta functions

  • Apéry's constant - the solution to ζ(3)
  • Artin conjecture
  • Basel problem boils down to ζ(2)
  • 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