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Dedekind Sums VOL AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS 16 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 16 Dedekind Sums Dedekind Sums Rademacher | Grosswald Noncommutative Rings provides a cross-section of ideas, techniques, and results that give the reader an idea of that part of algebra which concerns itself with noncommutative rings. In the space of 200 pages, Dedekind Sums Herstein covers the Jacobson radical, semisimple rings, commuta- tivity theorems, simple algebras, representations of nite groups, polynomial identities, Goldie's theorem, and the Golod–Shafarevitch theorem. Almost every practicing ring theorist has studied portions of Hans Rademacher this classic monograph. and Herstein's book is a guided tour through a gallery of masterpieces. The author's style is always elegant and his proofs always enlight- Emil Grosswald ening… I had a lot of pleasure when I rst read this book while I was an undergraduate student attending to a course given by C. Procesi at the University of Rome. Today, I appreciate even more the author's mastery and real gift for exposition. —Fabio Mainardi, MAA Reviews AMS / MAA PRESS 4-color Process 119 pages spine: 1/4" finish size: 5.5" X 8.5" 50 lb stock 10.1090/car/016 DEDEKIND SUMS By HANS RADEMACHER and EMIL GROSSWALD THE CARUS MATHEMATICAL MONOGRAPHS Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications IVAN NIVEN, Chairman Subcommittee on Carus Monographs R. G. BARTLE, Chairman R. P. BOAS D. T. FINKBEINER II HE CARUS MATHEMATICAL MONOGRAPHS are an expression of the desire of Mrs. Mary Hegeler Carus, and of T her son, Dr. Edward H. Carus, to contribute to the dissemination of mathematical knowledge by making accessible at nominal cost a series of expository presentations of the best thoughts and keenest researches in pure and applied mathematics. The publication of the first four of these monographs was made possible by a notable gift to the Mathematical Association of America by Mrs. Carus as sole trustee of the Edward C. Hegeler Trust fund. The sales from these have resulted in the Carus Monograph Fund, and the Mathematical Association has used this as a revolving book fund to publish the succeeding monographs. The exposition of mathematical subjects which the monographs contain are set forth in a manner comprehensible not only to teachers and students specializing in mathematics, but also to scientific workers in other fields, and especially to the wide circle of thoughtful people who, having a moderate acquaintance with elementary mathematics, wish to extend their knowledge without prolonged and critical study of the mathematical journals and treatises. The scope of this series includes also historical and biographical monographs. The following monographs have been published: No. 1. Calculus of Variations, by G. A. BLISS No. 2. Analytic Functions of a Complex Variable, by D. R. CURTISS No. 3. Mathematical Statistics, by H. L. RIETZ No. 4. Projective Geometry, by J. W. YOUNG No. 5. A History of Mathematics in America before 1900, by D. E. SMITH and JEKUTHIEL GINSBURG (out of print) No. 6. Fourier Series and Orthogonal Polynomials, by DUNHAM JACKSON No. 7. Vectors and Matrices, by C. C. MACDUFFEE No. 8. Rings and Ideals, by Ν. H. MCCOY No. 9. The Theory of Algebraic Numbers, by HARRY POLLARD No. 10. The Arithmetic Theory of Quadratic Forms, by B. W. JONES No. II. Irrational Numbers, by IVAN NIVEN No. 12. Statistical Independence in Probability, Analysis and Number Theory, by MARK KAC No. 13. A Primer of Real Functions, by RALPH P. BOAS, JR. No. 14. Combinatorial Mathematics, by HERBERT JOHN RYSER No. 15. Noncommutative Rings, by I. N. HERSTEIN No. 16. Dedekind Sums, by HANS RADEMACHER and EMIL GROSSWALD iv The Carus Mathematical Monographs NUMBER SIXTEEN DEDEKIND SUMS By HANS RADEMACHER, Late Professor of Mathematics University of Pennsylvania and Rockefeller University and EMIL GROSSWALD Professor of Mathematics Temple University Published and Distributed by THE MATHEMATICAL ASSOCIATION OF AMERICA Copyright © 1972 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Number: 72-88698 Paperback ISBN 978-0-88385-048-0 eISBN 978-1-61444-016-1 Hardcover (out of print) ISBN 978-0-88385-016-9 Printed in the United States of America. INTRODUCTION Professor Hans Rademacher was invited by the Mathe- matical Association of America to deliver the Earle Raymond Hedrick lectures at the 1963 summer meeting in Boulder, Colorado. Professor Rademacher chose the topic Dedekind Sums, and prepared a set of notes from which to deliver the lectures. However, a temporary illness prevented him from giving the lectures, and he prevailed upon his colleague and former student, Emil Grosswald, to make the presentation from the notes. Professor Rademacher never edited for publication the notes he had prepared for the Hedrick lectures. However, after his death in 1969, the manuscript was found among his papers, with a signed request that Emil Grosswald edit and publish these lecture notes. Professor Grosswald responded affirmatively, and completed the editing of these somewhat fragmentary notes, which consisted of 45 hand- written pages and a sketch of a bibliography. In view of the extensive additions of proofs, historical remarks, and subsequent developments of the subject, made by Emil Grosswald, the Subcommittee on Carus Monographs of the Committee on Publications concluded that joint authorship of the finished monograph was appropriate. The Association is indebted to Professor Grosswald for his dedication in bringing this volume into print. IVAN NIVEN, Chairman of the Committee on Publications vii PREFACE The Mathematical Association of America nominated Professor Hans Rademacher as Hedrick Lecturer for the summer meeting of 1963. As topics for these lectures, Professor Rademacher selected the "Dedekind Sums", a subject to which he had returned many times throughout his long and distinguished career and to which he had con- tributed immensely. He prepared a set of notes (called in what follows the "Notes"), but a passing indisposition prevented him from delivering the lectures. He recovered soon afterwards, was once more active in mathematics and wrote at least seven papers (among which there is also one on Dedekind sums) after 1963, but, for unknown reasons, failed to edit for publication his "Notes" of the Hedrick Lectures. In September of 1967, Professor Rademacher was stricken by a cruel illness, from which he never recovered. After his death on February 7, 1969, the manuscript was found among his papers; on the first page had been added the following words, in Professor Rademacher's handwriting: "If 1 should be unable to publish these lectures, I wish to ask Emil Grosswald to edit and publish them, (signed) Hans Rademacher (dated) 8-th September, 1963." Having had the privilege to be first Professor Rademacher's student, then, for many years, his colleague at the University of Pennsylvania and — I dare hope at least—his friend, ix χ PREFACE there could be no question on my part about the acceptance of this assignment. This turned out to be more difficult than anticipated. The "Notes" consist of 45 handwritten pages and a sketch (2 pages) of a bibliography. The text is written in that specific, personal style, which defies imitation and makes anyone who had ever attended his lectures believe that, while reading, he actually hears the familiar voice of that great teacher. Most proofs are suppressed — which is the reasonable thing for a Hedrick Lecture; instead, where a proof should appear, there is usually a reference to some paper containing it. The corresponding proof may have been used (either as actually published, or, more likely, as modified and stream- lined by Professor Rademacher) in editing the "Notes" for publication. These references are often rather cryptic, such as [Rdm], or [Iseki], or even just [ ], [ ]. He knew, but we must guess which of the 3, 4, or more different proofs of Rademacher he had in mind, or what paper of which Iseki is meant (there are three active mathematicians of this name and two of them work on topics germane to the present one), or which of the several existing proofs by different mathematicians was to be used at a given place. What was I supposed to do? It was impossible to take liberties with the text without risking to destroy what I consider one of its most valuable assets, its own characteristic style. On the other hand, it seemed indispensable to "flesh out" the "Notes" meant for oral presentation, by incorpor- ating into them at least some proofs, some indications of the history of the subject matter and the impact it has had on subsequent developments. After long hesitations and consultations with a referee and with Professor R. G. Bartle, Editor of the Carus Mono- PREFACE xi graphs, I decided on a compromise. I left the text of the "Notes" virtually unchanged, except for minor modifications required for clarity or for reasons of grammar*. Concerning the proofs that were missing and seemed desirable, in most cases I expanded existing, sketchy indications into proofs, if that could be done without overly long interruptions of the original text. In the other cases, I wrote up the proofs to the best of my ability, by making use to the largest possible extent of Professor Rademacher's own published papers. These proofs are collected into an Appendix and may be skipped at a first reading without impairing the clarity or continuity of the main text. I also wrote up a few pages on the history of the Dedekind sums and inserted these as a sixth Chapter. It is in the nature of such a brief survey that not all valu- able contributions to the theory of Dedekind sums could be mentioned. I have tried to select a representative sample, but wish to apologize to all those mathematicians who have contributed to this field and whose work is not mentioned here.
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