Topics in Multiplicative Number Theory
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'2'27 Hugh L. Montgomery
Topics in Multiplicative Number Theory
Springer-Verlag Berlin Heidelberq New York Tokyo
Author Hugh L. Montgomery Mathematics Department, University of Michigan Ann Arbor, MI 48109, USA
1st Edition 1971 2nd Printing 1986
Mathematics Subject Classification (1980): 10Hxx, 10J 15 ISBN 3-540-05641-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-05641-6 Springer-Verlag New York Heidelberg Berlin Tokyo
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1971 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PRE F ACE
These Notes are designed to present a survey of the present state of knowledge concerning those subjects touched upon in the last fifty pages of Davenport's MUltiplicative Number Theory.
Davenport's book is an
admirable introduction to the subject at hand; most results prerequisite to these Notes are found in
§§ 1-22
of his book. The more general results in this area are found in the first chapters.
Applications are then made to the
zeta-function and L-functions, and finally these results are used to derive theorems concerning the distribution of prime numbers. Due to continuing research in this field, these Notes are already a little out of date.
In particular, Jutila
has sharpened my estimate (12.14) in Theorem 12.2 and I am now in a position to say more concerning my Conjecture
9.2. I am indebted to many people for their suggestions and remarks.
In addition, E. Bombieri, P. X. Gallagher,
I
I
G. Halasz, M. H. Huxley, A. Selberg, P. Turan and R. Vaughan have made available to me valuable unpublished material.
My presentation has benefited
from the specific
suggestions and criticisms of Dr. Baker, Professor Bateman, and Professor Davenport.
While researching and writing
I have been grateful to receive monetary support from the Marshall Aid Commemoration Commission (London), the
IV National Science Foundation (Washington), Trinity College (Cambridge), the Institute for Advanced Study (Princeton', and the Air Force Office of Scientific Research (Washington) My thanks also go to Mrs. J. M. Jones for her diligent typing of my manuscript.
H.
L. M.
LOG I CAL
S T R UC T UR E
Solid lines indicate that some result in the later chapter depends on some result in the earlier chapter. Chapters 2,
5, 6
share common ideas, but the results are
not interdependent.
Lemma 17.3, which is quoted from the
literature, could be proved in such a manner that it would depend on the results of Chapter 16.
NOT A T ION
i,
We employ the familiar notations O( .f}.(
)•
./l:t (
i
, then we write f
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