Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function. The Riemann zeta function 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 it in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.
Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series
where I ranges through the non-zero ideals of the ring of integers OK of K and NK/Q(I) denotes the absolute norm of I (which is equal to both the index [OK : I] of I in OK or equivalently the cardinality of quotient ring OK / I). This sum converges absolutely for all complex numbers s with real part Re(s) > 1. In the case K = Q, this definition reduces to that of the Riemann zeta function.
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