Graduate Texts in Mathematics 206 Readings in Mathematics s Axler Editorial Board F.W Gehring K.A Ribet Graduate Texts in Mathematics Readings in Mathematics EbbinghausJHennesfHirzebruchIKoecherlMainzerlNeukirchIPrestelJRemmert: Numbers FultonJHarris: Representation Theory: A First Course Murty: Problems in Analytic Number Theory Remmert: Theory o/Complex Functions Walter: Ordinary Differential Equations Undergraduate Texts in Mathematics Readings in Mathematics Anglin: Mathematics: A Concise History and Philosophy Anglin!Lambek: The Heritage o/Thales Bressoud: Second Year Calculus HairerlWanner: Analysis by Its History HänunerlinJHoffrnann: Numerical Mathematics Isaac: The Pleasures 0/ Probability LaubenbacherlPengelley: Mathematical Expeditions: Chronicles by the Explorers Samuel: Projective Geometry Stillweil: Numbers and Geometry Toth: Glimpses 0/Algebra and Geometry M Ram Murty Problems in Analytic Number Theory Springer M RamMurty Department of Mathematics Queen's University Kingston, Ontario K7L 3N6 Canada Editorial Board S ruder F.W Gehring Mathematics Department East Hall University of Michigan Ann Arbor, M148109 USA Mathematics Department San Francisco State University San Francisco, CA 94132 USA K.A Ribet Mathematics Department University of California at Berke1ey Berkeley, CA 94720-3840 USA Cover photo by Dr C.J Mozzochi Mathematics Subject Classification (2000): 11Mxx, IlNxx Library of Congress Cataloging-in-Publication Data Murty, Maruti Ram Problems in analytic nurnber theory I M Ram Murty p cm - (Graduate texts in mathematics ; 206) Includes bibliographical references and index ISBN 978-1-4757-3443-0 ISBN 978-1-4757-3441-6 (eBook) DOlI0.I007/978-1-4757-3441-6 Number theory QA241 M87 2000 512'.73-dc21 I Title Printed on acid-free paper 11 Series 00-061865 2001 Springer-Verlag New York, Inc © 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc in 2001 Softcover reprint of the hardcover st edition 2001 All rights reserved This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrievaI, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be laken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone Production managed by Jenny Wolkowicki; manufacturing supervised by Jerome Basma Camera-ready copy provided by the author 987 543 I SPIN 10780694 Like fire in a piece of flint, knowledge exists in the mind Suggestion is the friction which brings it out - Vivekananda Preface "In order to become proficient in mathematics, or in any subject," writes Andre Weil, "the student must realize that most topics involve only a small number of basic ideas." After learning these basic concepts and theorems, the student should "drill in routine exercises, by which the necessary reflexes in handling such concepts may be acquired There can be no real understanding of the basic concepts of a mathematical theory without an ability to use them intelligently and apply them to specific problems." Weil's insightfulobservation becomes especially important at the graduate and research level It is the viewpoint of this book Our goal is to acquaint the student with the methods of analytic number theory as rapidly as possible through examples and exercises Any landmark theorem opens up a method of attacking other problems Unless the student is able to sift out from the mass of theory the underlying techniques, his or her understanding will only be academic and not that of a participant in research The prime number theorem has given rise to the rich Tauberian theory and a general method of Dirichlet series with which one can study the asymptotics of sequences It has also motivated the development of sieve methods We focus on this theme in the book We also touch upon the emerging Selberg theory (in Chapter 8) and p-adic analytic number theory (in Chapter 10) viii Preface This book is a collection of about five hundred problems in analytic number theory with the singular purpose of training the beginning graduate student in some of its significant techniques As such, it is expected that the student has had at least a semester course in each ofreal and complex analysis The problems have been organized with the purpose of self-instruction Those who exercise their mental muscles by grappling with these problems on a daily basis will develop not only a knowledge of analytic number theory but also the discipline needed for self-instruction, which is indispensable at the research level The book is ideal for a first course in analytic number theory either at the senior undergraduate level or the graduate level There are several ways to give such a course An introductory course at the senior undergraduate level can focus on chapters 1, 2, 3, 9, and 10 A beginning graduate course can in addition cover chapters 4, 5, and An intense graduate course can easily cover the entire text in one semester, relegating some of the routine chapters such as chapters 6, 7, and 10 to student presentations Or one can take up a chapter a week during a semester course with the instructor focusing on the main theorems and illustrating them with a few worked examples In the course of training students for graduate research, I found it tedious to keep repeating the cyclic pattern of courses in analytic and algebraic number theory This book, along with my other book "Problems in Algebraic Number Theory" (written jointly with J Esmonde), which appears as Graduate Texts in Mathematics, Vol 190, are intended to enable the student gain a quick initiation into the beautiful subject of number theory No doubt, many important topics have been left out Nevertheless, the material included here is a "basic tool kit" for the number theorist and so me of the harder exercises reveal the subtle "tricks of the trade." U nless the mi nd is challenged, it does not perform The student is therefore advised to work through the quest ions with some attention to the time factor "Work expands to fill the time allotted to it" and so if no upper limit is assigned, the mind does not get focused There is no universal rule on how long one should work on a problem However, it is a well-known fact that self-discipline, whatever shape it may take, opens the door for inspiration If the mental muscles are exercised in this fashion, the nuances of the solution become clearer and significant In this way, it is hoped that many, who not have Preface IX access to an "extern al teacher" will benefit by the approach of this text and awaken their "internal teacher." Princeton, November 1999 M Ram Murty Acknowledgments I would like to thank Roman Smirnov for his excellent job of typesetting this book into Jb'IEX I also thank Amir Akbary, Kalyan Chakraborty, Alina Cojocaru, Wentang Kuo, Yu-Ru Liu, Kumar Murty, and Yiannis Pe tri dis for their comments on an earlier version of the manuscript The text matured from courses given at Queen's University, Brown University, and the Mehta Research Institute I thank the students who participated in these courses Since it was completed while the author was at the Institute for Advanced Study in the fall of 1999, I thank lAS for providing a congenial atmosphere for the work I am grateful to the Canada Council for their award of a Killam Research Fellowship, which enabled me to devote time to complet,e this project Princeton, November 1999 M Ram Murty 442 10 p-adic Methods and the result is now clear D 10.5.2 With cjJ as in the previous exercise, show that cjJ(a + pt) : : : : cjJ(a) - at (modp), where aa::::::: (modp) We have (a + pt)p-l aP- + p(p - 1)taP- (mod p2) + pcjJ(a) - ptaP-1a (mod p2) + pcjJ(a) - pt(l + pcjJ(a))a (mod p2) + pcjJ(a) - pta (mod p2), from which the congruence follows D 10.5.3 Let [xl denote the greatest integer less than or equal to x For ~ a ~ p - 1, show that aP - a [aj] ::::::: L :- p J P p-l (modp) j=l We have p-l L p-l p-l j=l j=l L cjJ(a) + L cjJ(j) (modp) ~(aj) j=l p-l (p - l)cjJ(a) +L cjJ(j) (modp) j=l Thus cjJ(a) : : : : Write aj = rj p-l p-l j=l j=l L cjJ(j) - L cjJ(aj) (modp) + pqj, where ::; rj ::; p - Then by Exercise 10.5.2, cjJ(aj) = cjJ(rj so that p-l + pqj) q : : : : cjJ(rj) - (modp), Tj p-l p-l j=l j=l L cjJ(aj) L cjJ(Tj) - L /; (modp) = j=l q J 10.5 Supplementary Problems 443 Clearly, as j runs through to p - 1, so does r j Hence p-1 L q"/ (modp) cjy(a) == j=l ) = [aj/p], so that Now, aj == rj (modp) and qj L -1:- [ aJ"] p-1 acjy(a) == j=l (modp) P J o as desired 10.5.4 Prove the following generalization of Wilson's theorem: (p - k)!(k - 1)! == (_1)k (modp) for :::; k :::; p - Write -1 == (p - 1)! == (p - 1) (p - 2) (p - (k - 1)) (p - k)! (mod p) == (_1)k-1(k -1)!(P - k)! (modp), o from which the result folIows 10.5.5 Prove that for an odd prime p, 2P- _ _ - = p Deduce that 2p - L p-1 (-l)j+1 2' (modp) J j=l == (mod p2) if and only if the numerator of 1 1 + ···_-2 p-1 is divisible by p We have, 2P- - p (1 + 1)P - = 2p p-1 (p - 1)! 2?=(P-j)!j" )=1 ~ L (~) p-1 2p")=1 J 444 10 p-adic Methods By Wilson's theorem the numerator of each summand is congruent to -1 (modp) By Exercise 10.5.4, the denominator is congruent to (-l)j j{modp) Thus 2P- _ 1_ L p-l = p j=1 (-l)jH 2· J (modp), o as desired 10.5.6 Let p be an odd prime Show that for all x E Zp, hp{x)rp{x), where if hp{x) = { -x -1 if Ixlp = r p{x + 1) = 1, Ixlp < From the definition, we have if (n,p) = 1, if (n,p) 1= o The result now follows by continuity 10.5.7 For s ~ 2, show that the only solutions of x are x == 1, _1,2 - - 1, and 28 - + == (mod2 8) We have 28 1{x -1) Since x -1 = (x -l)(x + 1), exactly one of (x - 1) or (x + 1) is divisible by Either 211(x -1) or 211(x + 1) In the former case, x == -1 (mod 28 - ), so that x = 28 - 1t - for some t If t is even, we get x == -1 (mod2 ) If t is odd, we get x == 28 - - (mod 28 ) In the latter case, x == (mod 28 - ), and if t is odd, we get x == 28 - + (mod 28 ) 10.5.8 (The 2-adic r-function) Show that the sequence defined by r2(n)=(-lt rr j l:::;j