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
• 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