In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
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 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
External links A directory of all known zeta functions