This page intentionhally left blank A PRIMER OF ANALYTIC NUMBER THEORY Thisundergraduateintroductiontoanalyticnumbertheorydevelopsanalytic skills in the course of a study of ancient questions on polygonal numbers, perfect numbers, and amicable pairs. The question of how the primes are distributedamongallintegersiscentralinanalyticnumbertheory.Thisdis- tribution is determined by the Riemann zeta function, and Riemann’s work showshowitisconnectedtothezerosofhisfunctionandthesignificanceof theRiemannHypothesis. Startingfromatraditionalcalculuscourseandassumingnocomplexanal- ysis, the author develops the basic ideas of elementary number theory. The textissupplementedbyaseriesofexercisestofurtherdeveloptheconcepts andincludesbriefsketchesofmoreadvancedideas,topresentcontemporary researchproblemsatalevelsuitableforundergraduates.Inadditiontoproofs, bothrigorousandheuristic,thebookincludesextensivegraphicsandtables tomakeanalyticconceptsasconcreteaspossible. Jeffrey Stopple is Professor of Mathematics at the University of California, SantaBarbara. A PRIMER OF ANALYTIC NUMBER THEORY From Pythagoras to Riemann JEFFREY STOPPLE UniversityofCalifornia,SantaBarbara    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , United Kingdom Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521813099 © Jeffrey Stopple 2003 This book is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 -  isbn-13 978-0-511-07316-8 eBook (EBL) -  isbn-10 0-511-07316-X eBook (EBL) -  isbn-13 978-0-521-81309-9 hardback -  isbn-10 0-521-81309-3 hardback isbn--13 978-0-521-01253-9 paperback -  isbn-10 0-521-01253-8 paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Thisbookisdedicatedtoalltheformerstudents wholetmepracticeonthem. Contents Preface pageix Chapter1. SumsandDifferences 1 Chapter2. ProductsandDivisibility 24 Chapter3. OrderofMagnitude 43 Chapter4. Averages 64 Interlude1. Calculus 83 Chapter5. Primes 96 Interlude2. Series 111 Chapter6. BaselProblem 146 Chapter7. Euler’sProduct 159 Interlude3. ComplexNumbers 187 Chapter8. TheRiemannZetaFunction 193 Chapter9. Symmetry 216 Chapter10. ExplicitFormula 229 Interlude4. ModularArithmetic 254 Chapter11. Pell’sEquation 260 Chapter12. EllipticCurves 274 Chapter13. AnalyticTheoryofAlgebraicNumbers 295 Solutions 327 Bibliography 375 Index 379 vii