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The Riemann zeta function ζ(z) plotted with domain coloring. [1] The pole at = and two zeros on the critical line.. The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (), is a mathematical function of a complex variable defined as () = = = + + + for >, and its analytic continuation elsewhere.
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 ...
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
Zeta functions and L-functions express important relations between the geometry of Riemann surfaces, number theory and dynamical systems.Zeta functions, and their generalizations such as the Selberg class S, are conjectured to have various important properties, including generalizations of the Riemann hypothesis and various relationships with automorphic forms as well as to the representations ...
Let f be a function defined on a manifold M, such that the set of fixed points Fix(f n) is finite for all n > 1. Further let φ be a function on M with values in d × d complex matrices. The zeta function of the first kind is [ 1 ]
The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold. One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.
In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as (,) = (= ())where V is a non-singular n-dimensional projective algebraic variety over the field F q with q elements and N k is the number of points of V defined over the finite field extension F q k of F q.
Upload file; Search. Search. Appearance. Donate; ... Download as PDF; Printable version; ... In mathematics, Matsumoto zeta functions are a type of zeta function ...