Probabilistic Number Theory I Mean-Value Theorems
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Editors
S. S. Chern J. L. Doob J. Douglas, jr. A. Grothendieck E. Heinz F. Hirzebruch E. Hopf S. Mac Lane W. Magnus M. M. Postnikov w. Schmidt D. S. Scott K. Stein J. Tits B. L. van der Waerden Managing Editors B. Eckmann J. K. Moser
P. Turan (1910-1976)
P. Erdos (1913-
M. Kac (1914-
J. Kubilius (1921-
P.D.T.A. Elliott
Probabilistic Number Theory I Mean-Value Theorems
Springer-Verlag New York Heidelberg Berlin
P.D.T.A. Elliott Department of Mathematics University of Colorado Boulder, Colorado 80309 USA
AMS Subject Classifications (1980): lOKXX, 60B99, 60F99 Library of Congress Cataloging in Publication Data
Elliott, Peter D Probabilistic number theory. (Grundlehren der mathematischen Wissenschaften; 239-240) Bibliography: p. Includes index. CONTENTS: v. 1. Mean-value theorems.-v. 2. Central limit theorems. 1. Probabilistic number theory. I. Title. II. Series. QA241.7.E55 512'.7 79-20824 All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag.
©
1979 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1979 9 8 7 654 3 2 1 ISBN-13:978-1-4612-9991-2 DOl: 10.1007/978-1-4612-9989-9
e-ISBN-13:978-1-4612-9989-9
This book is dedicated to: Arthur George Elliott, my father, Martha Chalk Elliott, nee Ralph, my mother, to the dark melody of life, with its firefly flashes, to the journey along the I1lathematical rainbow, and to the memory of E. T. A. HoffI1lann, who would have appreciated the irony.
Acknowledgments
The author would like to thank Thomas Y. Crowell for permission to reproduce an extract from G. Marek's book Gentle Genius, and Albert Blanchard for permission to reproduce (in translation) a passage from Quelques aspects de la pensee d' un mathematicien, by P. Levy. I would also like to thank The Pacific Journal of Mathematics for allowing me to reproduce part of the paper On the distribution of numbers of the form O"(n)/n and some related questions, by P. Erdos, which appeared in volume 52 of their 1974 issue. Detailed references are given at the appropriate places. It has been my very good fortune to correspond with P. Erdos. M. Kac, J. Kubilius, and the late P. Tunin, who played important roles in the foundation and development of the Probabilistic Theory of Numbers. I would like to thank them for their permission to reproduce some of this correspondence which is of historical as well as mathematical interest. I would also like to thank P. Erdos and C. Ryavec for reading and commenting upon the manuscript. I thank Janice Wilson who, in the summer of 1977 , expertly typed the bulk of the manuscript. My thanks also go to the National Science Foundation of the United States for support on contracts numbers GP33026X, MCS75-08233 and MCS78-04374. Last, but not least, I would like to thank my wife Jean for her continued interest and support, and for sailing serenely through the sea of yellow pages with which I for several years covered the floor of our apartment. Boulder, Colorado August 197
