In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q. In particular, it can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.
The Dedekind zeta function is named for Richard Dedekind who introduced them in his supplement to P.G.L. Dirichlet's Vorlesungen über Zahlentheorie.
Read more about Dedekind Zeta Function: Definition and Basic Properties, Special Values, Relations To Other L-functions, Arithmetically Equivalent Fields
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