The Development of Prime Number Theory From Euclid to Hardy and Litt
1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's Elemen ta, where we find a proof of their infinitude, now regarded as canonical. One feels that Euclid's a
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Springer-Verlag Berlin Heidelberg GmbH
Wladyslaw Narkiewicz
The Development of Prime Number Theory From Euclid to Hardy and Littlewood
i
Springer
Wladyslaw Narkiewicz Institute of Mathematics Wrodaw University Plac Grunwaldzki 2-4 50384 Wrodaw, Poland e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data applied for Die DeuIscbe BibliOlbek - CIP-Binbeitsaufilalune
Narkiewicz, WladysJaw:
1be dcvelopment of prime number thcory : rrom Euelid lO Hardy II1II Lilllewood I Wlldyslaw Narkiewicz. (Springer Monograpbs ia mathcmatics)
ISBN 978-30642-085574 ISBN 978-3-662-13157-2 (eBook) DOI 10.1007/978-3-662-13157-2
Mathematics Subject C1assification (1991): 1l-03, llA41, llMxx, llNOS,llN13,llN25
ISSN 1439-7382
ISBN 978-3-642-08557-4 This work is subject ta copyright All rights are reserved, whether the whole ar part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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 Berlin HeideIbeIg GmbIl. Violations are liable for prosecution under the German Copyright Law.
o Springer-Verlag Berlin Heidelberg 2000
Originally publisbed by Springer-Verlag Berlin Heidelberg New York in 2000 Softcover reprint of the hardcover l st edition 2000
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective 1aws and regulations and therefore free for general use. Printed an acid-free paper 44/3111 - 5432
Preface
1. People were already interested in prime numbers in ancient times, and the first result concerning the distribution of primes appears in Euclid's Elementa, where we find a proof of their infinitude, now regarded as canonical. One feels that Euclid's argument has its place in The Book, often quoted by the late Paul ErdOs, where the ultimate forms of mathematical arguments are preserved. Proofs of most other results on prime number distribution seem to be still far away from their optimal form and the aim of this book is to present the development of methods with which such problems were attacked in the course of time. This is not a historical book since we refrain from giving biographical details of the people who have played a role in this development and we do not discuss the questions concerning why each particular person became interested in primes, because, usually, exact answers to them are impossible to obtain. Our idea is to present the development of the theory of the distribution of prime numbers in the period starting in antiquity and concluding at the end of the first decade of the 20th century. We shall also present some later developments, mostly in s
