Analytic Arithmetic in Algebraic Number Fields
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1205 B.Z. Moroz
Analytic Arithmetic in Algebraic Number Fields
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
B.Z. Moroz Max-Planck-Institut fur Mathematik, Universitiit Bonn Gottfried-Claren-Str. 26,5300 Bonn 3, Federal Republic of Germany
Mathematics Subject Classification (1980): 11057, 11R39, 11R42, 11R44, 11R45, 22C05 ISBN 3-540-16784-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-16784-6 Springer-Verlag New York Berlin Heidelberg
Library of Congress Cataloging-in-Publication Data. Moroz, B. Z. Analytic arithmetic in algebraic number fields. (Lecture notes in mathematics; 1205) "Subseries: Mathematisches Institut der Universitat und Max-Planck-Institut fur Mathematik. Bonn - vol, 7." Bibliography: p. Includes index. 1. Algebraic number theory. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag; 1205. QA3.L28 no. 1205 [0A247] 510 [512'.74] 86·20335 ISBN 0-387-16784-6 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations. broadcasting. reproduction by photocopying machine or similar means. and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort". Munich.
© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
Introduction. This book is an improved version of our memoir that appeared in Bonner Mathematische Schriften, [64].
Its purpose is twofold: first, we give
a complete relatively self-contained proof of the theorem concerning analytic continuation and natural boundary
of Euler products (sketched
in Chapter III of [64]) and describe applications of Dirichlet series represented by Euler products under consideration; secondly, we review in detail classical methods of analytic number theory in fields of algebraic numbers.
Our presentation of these methods (see Chapter I) has
been most influenced by the work of E. Landau, [24], and A. Weil,
[91]
(cf. also [87]).
[40], [42], E. Heeke,
In Chapter II we develop
formalism of Euler products generated by polynomials whose coefficients lie in the ring of virtual characters of the (absolute) Weil group of a number field and apply it to study scalar products of Artin-Weil Lfunctions.
This leads, in particular, to a solution of a long-standing
problem concerning analytic behaviour of the scalar products, or convolutions, of L-functions Hecke "mit Grossencharakteren" (cf , [63] for the history of this problem; one may regard this note as a resume of Chapter II, if you like).
Chapter III describes applications of those
scalar products to the problem of asymptotic distribution of integral and prime ideals having equal norms and to a classical problem about distribution of integral points on a variety defined by a system of norm-forms.
Chapter IV is designed to relate the contents
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