Selberg's Zeta-, L-, and Eisensteinseries
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Ulrich Christian
Selberg's Zeta-, L:, and Eisensteinseries
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Author
Ulrich Christian Mathematisches Institut, Georq-Auqust-Universitat Bunsenstr. 3-5, 3400 Gottinqen, Federal Republic of Germany
CR Subject Classifications (1982): 3,10 AMS Subject Classifications (1980): 10005,10020,10024 ISBN 3-540-12701-1 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12701-1 Springer-Verlag New York Heidelberg Berlin Tokyo
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.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
This course of lectures was given at the University of Gottingen in the summer-semester 1983. I thank Mrs. Christiane Gieseking for her careful typing of the troublesome manuscript.
Ulrich Christian
CONTENTS Introduction Chapter I. Epstein's zetafunction of a binary quadratic form § 1. Preliminaries § 2. Epstein's zetafunctions and L-series § 3. Elementary Eisenstein series
1 1
11
24
Chapter II. Preparational material § 4. Systems of primitive characters § 5. Matrices § 6. The Riemannian space of positive matrices § 7. Theta functions
28 29 36 53 66
Chapter III. Selberg's zeta- and L-series § 8. Descending chains § 9. Characters § 10. Selberg's zeta- and L-series § 11. Analytic continuation § 12. Functional equations § 13. Residues of Selberg's zetafunctions
89 89 98 102 116 137 139
Chapter IV. Selberg's Eisensteinseries § 14. Siegel's upper half-plane § 15. Selberg's Eisensteinseries § 16. Representation with Siegel's Eisensteinseries § 17. Representation with Selberg's zetafunction § 18. Analytic continuation
149 149 155 160 171 177
Chapter V. Siegel's Eisensteinseries § 19. Siegel's Eisensteinseries § 20. Poles and Hecke's summation
181 181 183
Literature List of symbols Index
187 191 195
INTRODUCTION In these lecture notes we prove analytic continuation and functional equations for Selberg's Eisensteinseries, Selberg's zetafunctions, Selberg's L-series, and Siegel's Eisensteinseries. We start with Epstein's zetafunction for a binary quadratic form and Epstein's L-functions which are connected with Epstein's zetafunction like Dirichlet's L-series are connected with Riemann's zetafunction. Then we consider Eisensteinseries for the elliptic modular group which are also closely related to Epstein's zetafunction. In the next chapters we come to Selberg's zetafunction (see [33J, § 17, Selberg [41J, and Terras [45J, [46J). Furthermore we consider Selberg's L-series which are connected with Selberg's zetafunctions like Dirichlet's L-series are conn
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