Basic Analysis of Regularized Series and Products
Analytic number theory and part of the spectral theory of operators (differential, pseudo-differential, elliptic, etc.) are being merged under amore general analytic theory of regularized products of certain sequences satisfying a few basic axioms. The mo
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F. Hirzebruch
1564
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, ZUrich F. Takens, Groningen Subseries: Mathematisches Institut der Universitat und Max-Planck-Institut fur Mathematik, Bonn - vol. 18 Advisor:
F. Hirzebruch
1564
Jay Jorgenson Serge Lang
Basic Analysis of Regularized Series and Products
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Authors Jay A. Jorgenson Serge Lang Department of Mathematics Yale University Box 2155 Yale Station New Haven, CT 06520, USA
Mathematics Subject Classification (l991): IIM35, IIM41, IIM99, 30B50, 30D15, 35P99,35S99,39B99,42A99 (Authors' Note: there is no MSC number for regularized products, but there should be.)
ISBN 3-540-57488-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57488-3 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 2146/3140-543210 - Printed on acid-free paper
Foreword The two papers contained in this volume provide results on which a series of subsequent papers will be based, starting with [JoL 92b], [JoL 92d] and [JoL 93]. Each of the two papers contains an introduction dealing at greater length with the mathematics involved. The two papers were first submitted in 1992 for publication in J. reine angew. Math. A referee emitted the opinion: "While such generalized products are of interest, they are not of such central interest as to justify a series of long papers in expensive journals." The referee was cautious, stating that this "view is subjective", and adding that he "will leave to the judgement of the editors whether to pass on this recommendation to the authors". The recommendation, in addition not to publish "in expensive journals", urged us to publish a monograph instead. In any case, the editors took full responsibility for the opinion about the publication of our series "in expensive journals". We disagree very strongly with this opinion. In fact, one of the applications of the complex analytic properties of regularized products contained in our first paper is to a generalization of Cramer's theorem, which we prove in great generality, and which appears in Math. Annalen [JoL 92b]. The referee for Math. Ann. characterized this result as "important and basic in the field" . Our papers were written in a selfcontained way, to provide a suitable background for an openended series. Thus we always considered the possible al
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