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Asymptotic probability measures of zeta-functions of algebraic number fields. (English) Zbl 0746.11051
Let \(K\) be an algebraic number field of degree \(l\) and \(\zeta_ K(s)\) its Dedekind zeta-function. The author considers the value distribution of \(\log \zeta_ K(s)\) on vertical lines in the half-plane \(\sigma > 1- L^{-1}\), where \(L=\max(l,2)\). It is shown that given a rectangle \(R\) in the complex plane and letting \(s\) run over a vertical line, \(\log \zeta_ K(s)\) lies in \(R\) with a certain asymptotic probability \(W(R)\). Moreover, the rate of convergence to this asymptotic probability is estimated.

MSC:
11R42 Zeta functions and \(L\)-functions of number fields
11M41 Other Dirichlet series and zeta functions
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
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