Explicit Formulas for Regularized Products and Series
The theory of explicit formulas for regularized products and series forms a natural continuation of the analytic theory developed in LNM 1564. These explicit formulas can be used to describe the quantitative behavior of various objects in analytic number
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1593
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. 21 Advisor: F. Hirzebruch
1593
Jay Jorgenson & Serge Lang Dorian Goldfeld
Explicit Formulas for Regularized Products and Series
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Authors Jay Jorgenson Serge Lang Mathematics Department Box 208283 Yale Station 10 Hillhouse Ave New Haven CT 06520-8283, USA Dorian Goldfeld Mathematics Department Columbia Unversity New York, NY 10027, USA
Mathematics Subject Classification (1991): 11 M35, 11M41, 11M99, 30B50, 30D 15, 35P99,35S99,42A99 Authors Note: there is no MSC number for regularized products, but there should be. ISBN 3-540-58673-3 Springer-Verlag Berlin Heidelberg New York CIP-Data applied for 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 1994 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10130182 46/3140-543210 - Printed on acid-free paper
EXPLICIT FORMULAS FOR REGULARIZED PRODUCTS AND SERIES
Jay Jorgenson and Serge Lang
A SPECTRAL INTERPRETATION OF WElL'S EXPLICIT FORMULA
Dorian Goldfeld
EXPLICIT FORMULAS FOR REGULARIZED PRODUCTS AND SERIES Jay Jorgenson and Serge Lang Introduction I
series 1. 2. 3. 4. 5. 6. 7.
estimates of regularized harmonic
Regularized products and harmonic series Asymptotics in vertical strips Asymptotics in sectors Asymptotics in a sequence to the left Asymptotics in a parallel strip Regularized product and series type Some examples
II Cramer's Theorem as an Explicit Formula 1. 2. 3. 4.
3 11 14 20 22 24 34 36 39
43
Euler sums and functional equations The general Cramer formula Proof of the Cramer theorem An inductive theorem
45 47 51 57
III Explicit Formulas under Fourier Assumptions
61
1. 2. 3. 4. 5.
Growth conditions on Fourier transforms The explicit formulas The terms with the q's The term involving cP The Wei I functional and regularized product type
62 66 73 78 79
IV From Functional Equations to Theta Inversions 85 1. An application of the explicit formulas 2. Some examples of theta inversions
87 92
VIII
V From Theta Inversions to Functional Equations 1. The Weil functional of a Gaussian test function 2. Gauss transforms 3. Theta inversions yield zeta functions 4. A new zeta function for compact quotients of M a
VI A Generalization of Fujii's Theorem 1.
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