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Arithmetical investigations of a certain infinite product

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Arithmetical investigations of a certain infinite product COMPOSITIO MATHEMATICA PETERBUNDSCHUH KEIJOVÄÄNÄNEN Arithmetical investigations of a certain infinite product Compositio Mathematica, tome 91, no 2 (1994), p. 175-199. <> © Foundation Compositio Mathematica, 1994, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: // implique l’accord avec les conditions gé- nérales d’utilisation ( 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 175 Arithmetical investigations of a certain infinite product PETER BUNDSCHUH1* and KEIJO VÄÄNÄNEN2 1 Mathematisches Institut der Un iversita t, Weyertal 86-90, D- W-5000 K’ôln 41, Germany; 2Matematiikan Laitos, Oulun Yliopisto, Linnanmaa, SF-90570 Oulu, Finland Received 6 October 1992; accepted in final form 5 April 1993 Compositio Mathematica 91: 175-199, 1994. © 1994 Kluwer Academic Publishers. Printed in the Netherlands. Introduction and main results Let K denote an algebraic number field of degree d over Q. For every place v of K we define d" : = [Kv: Qv]. If a finite place v of K lies over the prime p, we write v 1 p, and for an infinite place v of K we write v oo . We normalize the absolute value 1. 1, by where a e Q and 1 . dénotes the ordinary absolute value in R or in C. Then, for any a E K ", we have the product formula The absolute height h(a) of a E K is defined by the formula and the absolute height h(a) of the vector a = t (a o, a 1 ) ~K2 by For the whole paper we suppose that q is some fixed element from K satisfying |q|v &#x3E; 1 for some fixed valuation v of K, and furthermore lql,, ~ 1 for all w oo. It

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