What is Zeta function: Definition and 111 Discussions

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




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{\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

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