This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

Published for the Eidgenossische Technische Hochschule Zurich

Carl Ludwig Siegel gave a course of lectures on the Geometry of Numbers at New York University during the academic year 1945-46, when there were hardly any books on the subject other than Minkowski's original one. This volume stems from Siegel's requirements of accuracy in detail, both in the text and in the illustrations, but involving no changes in the structure and style of the lectures as originally delivered. This book is an enticing introduction to Minkowski's great work. It also reveals the workings of a remarkable mind, such as Siegel's with its precision and power and aesthetic charm. It is of interest to the aspiring as well as the established mathematician, with its unique blend of arithmetic, algebra, geometry, and analysis, and its easy readability.

From the Preface by K. Chandrasekharan: "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL. In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time...this publication...will no doubt stimulate generations of scholars to come." Volume III collects Siegel's papers from 1945 to 1964.

From the Preface by K. Chandrasekharan: "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL... In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time... this publication...will no doubt stimulate generations of scholars to come." Volume IV collects Siegels papers from 1968 to 1975.

From the Preface by K. Chandrasekharan: "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL, who is rising seventy. In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time...this publication...will no doubt stimulate generations of scholars to come." Volume II includes Siegel's papers written between 1937 and 1944.

From the Preface: "The name of Hermann Weyl is enshrined in the history of mathematics. A thinker of exceptional depth, and a creator of ideas, Weyl possessed an intellect which ranged far and wide over the realm of mathematics and beyond. His mind was sharp and quick, his vision clear and penetrating. Whatever he touched he adorned. His personality was suffused with humanity and compassion and a keen aesthetic sensibility. Its fullness radiated charm. He was young at heart to the end. By precept and example, he inspired many mathematicians and influenced their lives. The force of his ideas has affected the course of science. He ranks among the few universalists of our time. This collection of papers is a tribute to his genius. It is intended as a service to the mathematical community....These papers will no doubt be a source of inspirations to scholars through the ages." Volume IV comprises 46 articles written between 1941 and 1953.

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

The plan of this book had its inception in a course of lectures on arithmetical functions given by me in the summer of 1964 at the Forschungsinstitut fUr Mathematik of the Swiss Federal Institute of Technology, Zurich, at the invitation of Professor Beno Eckmann. My Introduction to Analytic Number Theory has appeared in the meanwhile, and this book may be looked upon as a sequel. It presupposes only a modicum of acquaintance with analysis and number theory. The arithmetical functions considered here are those associated with the distribution of prime numbers, as well as the partition function and the divisor function. Some of the problems posed by their asymptotic behaviour form the theme. They afford a glimpse of the variety of analytical methods used in the theory, and of the variety of problems that await solution. I owe a debt of gratitude to Professor Carl Ludwig Siegel, who has read the book in manuscript and given me the benefit of his criticism. I have improved the text in several places in response to his comments. I must thank Professor Raghavan Narasimhan for many stimulating discussions, and Mr. Henri Joris for the valuable assistance he has given me in checking the manuscript and correcting the proofs. K. Chandrasekharan July 1970 Contents Chapter I The prime number theorem and Selberg's method 1. Selberg's fonnula . . . . . . 1 2. A variant of Selberg's formula 6 12 3. Wirsing's inequality . . . . . 17 4. The prime number theorem. ."

Carl Ludwig Siegel gave a course of lectures on the Geometry of Numbers at New York University during the academic year 1945-46, when there were hardly any books on the subject other than Minkowski's original one. This volume stems from Siegel's requirements of accuracy in detail, both in the text and in the illustrations, but involving no changes in the structure and style of the lectures as originally delivered. This book is an enticing introduction to Minkowski's great work. It also reveals the workings of a remarkable mind, such as Siegel's with its precision and power and aesthetic charm. It is of interest to the aspiring as well as the established mathematician, with its unique blend of arithmetic, algebra, geometry, and analysis, and its easy readability.

In gratefuZ remerribrance of Marston Morse and John von Neumann This text formed the basis of an optional course of lectures I gave in German at the Swiss Federal Institute of Technology (ETH), Zlirich, during the Wintersemester of 1986-87, to undergraduates whose interests were rather mixed, and who were supposed, in general, to be acquainted with only the rudiments of real and complex analysis. The choice of material and the treatment were linked to that supposition. The idea of publishing this originated with Dr. Joachim Heinze of Springer Verlag. I have, in response, checked the text once more, and added some notes and references. My warm thanks go to Professor Raghavan Narasimhan and to Dr. Albert Stadler, for their helpful and careful scrutiny of the manuscript, which resulted in the removal of some obscurities, and to Springer-Verlag for their courtesy and cooperation. I have to thank Dr. Stadler also for his assistance with the diagrams and with the proof-reading. Zlirich, September, 1987 K. C. Contents Chapter I. Fourier transforms on L (-oo, oo) 1 1. Basic properties and examples . . . . . . . . . . . . . . . . 1 2. The L 1-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3. Differentiabili ty properties . . . . . . . . . . . . . . 18 4. Localization, Mellin transforms . . . . . . . . . . . . . . . . . . . . . . . 25 5. Fourier series and Poisson's summation formula . . . . . . . . . . 32 6. The uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

The description for this book, Fourier Transforms. (AM-19), will be forthcoming.

From the Preface (K. Chandrasekharan, 1966): "The publication of this collection of papers is intended as a service to the mathematical community, as well as a tribute to the genius of CARL LUDWIG SIEGEL, who is rising seventy.In the wide range of his interests, in his capacity to uncover, to attack, and to subdue problems of great significance and difficulty, in his invention of new concepts and ideas, in his technical prowess, and in the consummate artistry of his presentation, SIEGEL resembles the classical figures of mathematics. In his combination of arithmetical, analytical, algebraical, and geometrical methods of investigation, and in his unerring instinct for the conceptual and structural, as distinct from the merely technical, aspects of any concrete problem, he represents the best type of modern mathematical thought. At once classical and modern, his work has profoundly influenced the mathematical culture of our time."Volume I includes Siegel's papers written between 1921 and 1937.

From the Preface: "The name of Hermann Weyl is enshrined in the history of mathematics. A thinker of exceptional depth, and a creator of ideas, Weyl possessed an intellect which ranged far and wide over the realm of mathematics, and beyond. His mind was sharp and quick, his vision clear and penetrating. Whatever he touched he adorned. His personality was suffused with humanity and compassion, and a keen aesthetic sensibility. Its fullness radiated charm. He was young at heart to the end. By precept and example, he inspired many mathematicians, and influenced their lives. The force of his ideas has affected the course of science. He ranks among the few universalists of our time. This collection of papers is a tribute to his genius. It is intended as a service to the mathematical community....These papers will no doubt be a source of inspirations to scholars through the ages." Volume II comprises 38 articles written between 1918 and 1926.

From the Preface: "The name of Hermann Weyl is enshrined in the history of mathematics. A thinker of exceptional depth, and a creator of ideas, Weyl possessed an intellect which ranged far and wide over the realm of mathematics, and beyond. His mind was sharp and quick, his vision clear and penetrating. Whatever he touched he adorned. His personality was suffused with humanity and compassion, and a keen aesthetic sensibility. Its fullness radiated charm. He was young at heart to the end. By precept and example, he inspired many mathematicians, and influenced their lives. The force of his ideas has affected the course of science. He ranks among the few universalists of our time. This collection of papers is a tribute to his genius. It is intended as a service to the mathematical community....These papers will no doubt be a source of inspirations to scholars through the ages." Volume III comprises 52 articles written between 1926 and 1940.

From the Preface: "The name of Hermann Weyl is enshrined in the history of mathematics. A thinker of exceptional depth, and a creator of ideas, Weyl possessed an intellect which ranged far and wide over the realm of mathematics, and beyond. His mind was sharp and quick, his vision clear and penetrating. Whatever he touched he adorned. His personality was suffused with humanity and compassion, and a keen aesthetic sensibility. Its fullness radiated charm. He was young at heart to the end. By precept and example, he inspired many mathematicians, and influenced their lives. The force of his ideas has affected the course of science. He ranks among the few universalists of our time. This collection of papers is a tribute to his genius. It is intended as a service to the mathematical community....These papers will no doubt be a source of inspirations to scholars through the ages." Volume I comprises 29 articles written between 1908 and 1917.

Ganz in Hermann Weyls bekannt klarer Darstellung geschrieben, gibt dieser Beitrag einen Bericht uber die Entstehung der grundlegenden Ideen, die der modernen Geometrie zugrunde liegen. Diese Schrift spiegelt in einzigartiger Weise Weyls mathematische Personlichkeit wider. Sie richtet sich an alle, die sich mit Fragen der Topologiegruppentheorie, Differentialgeometrie und mathematischer Physik beschaftigen. From the foreword of the editor K. Chandrasekharan: "Written in Weyl's finest style, while he was rising forty, the article is an authentic report on the genesis and evolution of those fundamental ideas that underlie the modern conception of geometry. Part I is on the continuum, and deals with analysis situs, imbeddings, and coverings. Part II is on structure, and deals with infinitesimal geometry in its many aspects, metric, conformal, affine, and projective; with the question of homogeneity, homogeneous spaces from the group-theoretical standpoint, the role of the metric field theories in physics, and the related problems of group theory. It is hoped that this article will be of interest to all those concerned with the growth and development of topology, group theory, differential geometry, geometric function theory, and mathematical physics. It bears the unmistakable imprint of Weyl's mathematical personality, and of his remarkable capacity to capture and delineate the transmutation of some of the nascent into the dominant ideas of the mathematics of our time".

Diese Arbeit ist eine Zusammenfassung der Vorlesung, die ich im Wintersemester 1965/66 in englischer Sprache an der E.T.H. gehalten habe. Herr J. Steinig hat sie sorgf itigst in der Vortragssprache abgefasst und ins Deutsche Gbertragen. Die Herren M. BrGhlmann, H. Leutwiler und U. Suter haben den deutschen Text freundlichst dur- gelesen und an seiner endgGltigen, stilgerechten Fassung mitgearbeitet. Ihnen allen gebGhrt mein Dank. K.C. Literaturverzeichnis 1. G.H. Hardy and E.M. Wright, "An Introduction to the Theory of Numbers," Clarendon Press, Oxford, 1954. 2. H. Rademacher, "Lectures on Elementary Number Theory," Blaisdell Publishing Company, 1964. 3. A.E. Ingham, "The Distribution of Prime Numbers," Cambridge University Press, 1932. 4. H. Weyl, "Ueber die Gleichverteilung von Zahlen mod. Eins," Math. Annalen 77, 313-352 (1916). 5. C.L. Siegel, "Ueber Gitterpunkte in Convexen K6rpern und ein damit zusammenh ngendes Extremalproblem," Acta Math. 65, 307-323 (1935). Inhaltsverzeichnis IQ Der Fundamentalsatz der elementaren Zahlentheorie. II. Kongruenzen. III. Die rationale Approximation einer irrationalen Zahl. Der Satz von Hurwitz IV. Quadratische Reste, und die Darstellbarkeit einer positiven ganzen Zahl als Summe von vier Quadraten. V Das quadratische Reziprozit tsgesetz. VI. Zahlentheoretische Funktionen und Gitterpunkte. VII. Der Satz von Chebychev ber die Verteilung der Primzahlen. VIII. Die Weylsche "Gleichverteilung von Zahlen mod I," und der Satz von Kronecker. IX. Der Satz von Minkowski ber Gitterpunkte in konvexen Bereichen. X. Der Dirichletsche Satz von den Primzahlen in einer arithmetischen Progression.