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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. 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, 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.
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 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . ."
Published for the Eidgenossische Technische Hochschule Zurich
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.
This book has grown out of a course of lectures I have given at the
Eidgenossische Technische Hochschule, Zurich. Notes of those
lectures, prepared for the most part by assistants, have appeared
in German. This book follows the same general plan as those notes,
though in style, and in text (for instance, Chapters III, V, VIII),
and in attention to detail, it is rather different. Its purpose is
to introduce the non-specialist to some of the fundamental results
in the theory of numbers, to show how analytical methods of proof
fit into the theory, and to prepare the ground for a subsequent
inquiry into deeper questions. It is pub lished in this series
because of the interest evinced by Professor Beno Eckmann. I have
to acknowledge my indebtedness to Professor Carl Ludwig Siegel, who
has read the book, both in manuscript and in print, and made a
number of valuable criticisms and suggestions. Professor Raghavan
Narasimhan has helped me, time and again, with illuminating
comments. Dr. Harold Diamond has read the proofs, and helped me to
remove obscurities. I have to thank them all. K.C."
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 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 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 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.
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