Introduction to Analytic Number 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
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H erausgegeben von
J. L. Doob . E. Heinz' F. Hirzebruch . E. Hopf· H. Hopf W. Maak . S. MacLane . W. Magnus· D. Mumford M. M. Postnikov· F. K. Schmidt· D. S. Scott· K. Stein
Geschaftsfohrende H erausgeber
B. Eckmann und B. L. van der Waerden
K. Chandrasekharan
Introduction to Analytic Number Theory
I Springer-Verlag New York Inc. 1968
Prof. Dr. K. Chandrasekharan Eidgeniissische Technische Hochschule Ziirich
Geschiiftsfiihrende Herausgeber:
Prof. Dr. B. Eckmann Eidgeniissiscbe Technische Hochschule Ziirich
Prof. Dr. B. L. van der Waerden Mathernatisches Institut der Universitiit Ziirich
ISBN-13: 978-3-642-46126-2 e-ISBN-13: 978-3-642-46124-8 DOl: 10.1007/978-3-642-46124-8 Aile Rechte vorbehalten. Kein Teil dieses Buches darf ohne schriftliche Genehrnigung des Springer·Verlages iibersetzt odeT in irgendeiner Form vervietnUtigt werden.
© by Springer-Verlag Berlin· Heidelberg 1968 Softcover reprint of the hardcover 1st edition 1968 Library of Congress Catalog Card Number 68-21990
Titel-Nr. 5131
Preface
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 show to some of the fundamental results in the theory of numbers, how analytical methods of proof fit into the theory, and to prepare the ground for a subsequent inquiry into deeper questions. It is published in this series because of the interest evinced by Professor Beno Eckmann.
to
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. August 1968
K.C.
Contents Chapter I The unique factorization theorem § 1. § 2. § 3. § 4. § 5. § 6.
Primes . . . . . . . . . . . . The unique factorization theorem. . . . . . . . . A second proof of Theorem 2 . . . . . . . . . . . Greatest common divisor and least common multiple Farey sequences . . . . The infinitude of primes. . . . . . . . . . . . .
I I 3 5 6 9
Chapter II Congruences § 1. Residue classes. . . . . . . . . . . . § 2. Theorems of Euler and of Fermat. . . . § 3. The number of solutions of a congruence
11
13
15
Chapter III Rational approximation of irrationals and Hurwitz's theorem § 1. § 2. § 3. § 4.
Approximation of irrationals . Sums of two squares . . Primes of the form 4k ± 1 Hurwitz's theorem . . .
18 20 21 22 Chapter IV
Quadratic residues and the representation of a number as a sum of four squares § 1. § 2. § 3. § 4.
The Legendre symbol. . . . . . . . Wilson's theorem and Euler's criterion Sums
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