List of zeta functions

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

$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}.$

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
• 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
• Ihara zeta-function of a graph
• Igusa zeta-function
• 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

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
• Birch and Swinnerton-Dyer conjecture
• Riemann hypothesis and the generalized Riemann hypothesis.
• Selberg class S

External links

• A directory of all known zeta functions