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The asymptotic behaviour of additive functions in algebraic number theory

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The asymptotic behaviour of additive functions in algebraic number theory COMPOSITIO MATHEMATICA J. P. M. DEKROON The asymptotic behaviour of additive functions in algebraic number theory Compositio Mathematica, tome 17 (1965-1966), p. 207-261. <http://www.numdam.org/item?id=CM_1965-1966__17__207_0> © Foundation Compositio Mathematica, 1965-1966, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 207 The asymptotic behaviour of additive functions in algebraic number theory by J. P. M. de Kroon CHAPTER 1 Introduction § 1. The problem The aim of this paper is to prove some generalizations of a theorem of Erdôs and Kac concerning the behaviour of an additive numbertheoretic function on the natural numbers [7]. The generalizations which we shall establish concern: 1. the set of the principal integral ideals of an arbitrary algebraic numberfield, 2. some sets of algebraic integers contained in an arbitrary but fixed algebraic numberfield, and 3. the ring of integral elements contained in an algebraic function- field. We state that we shall formulate the problem in a rather general way, but only for the case where sets of algebraic integers are considered. It may be noted tha,t the formulation of the two other cases takes almost the same form. Before formulating the problem in question, it is useful to make some preliminary remarks with regard to certain definitions and notations in order to make it possible to give a short and comprehensible formulation of the problem. Let Q be the f

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