Problems in Analytic Number Theory

505 Pages·2007·8.91 MB·English

by  M. Ram Murty

206
Graduate Texts in Mathematics
EditorialBoard
S.Axler K.A.Ribet
GraduateTextsinMathematics
1 TAKEUTI/ZARING.Introductionto 36 KELLEY/NAMIOKAetal.Linear
AxiomaticSetTheory.2nded. TopologicalSpaces.
2 OXTOBY.MeasureandCategory.2nded. 37 MONK.MathematicalLogic.
3 SCHAEFER.TopologicalVectorSpaces. 38 GRAUERT/FRITZSCHE.SeveralComplex
2nded. Variables.
4 HILTON/STAMMBACH.ACoursein 39 ARVESON.AnInvitationtoC-Algebras.
HomologicalAlgebra.2nded. 40 KEMENY/SNELL/KNAPP.Denumerable
5 MACLANE.CategoriesfortheWorking MarkovChains.2nded.
Mathematician.2nded. 41 APOSTOL.ModularFunctionsand
6 HUGHES/PIPER.ProjectivePlanes. DirichletSeriesinNumberTheory.2nded.
7 J.-P.Serre.ACourseinArithmetic. 42 J.-P.SERRE.LinearRepresentationsof
8 TAKEUTI/ZARING.AxiomaticSetTheory. FiniteGroups.
9 HUMPHREYS.IntroductiontoLie 43 GILLMAN/JERISON.RingsofContinuous
AlgebrasandRepresentationTheory. Functions.
10 COHEN.ACourseinSimpleHomotopy 44 KENDIG.ElementaryAlgebraicGeometry.
Theory. 45 LOE`ve.ProbabilityTheoryI.4thed.
11 CONWAY.FunctionsofOneComplex 46 LOE`ve.ProbabilityTheoryII.4thed.
VariableI.2nded. 47 MOISE.GeometricTopologyin
12 BEALS.AdvancedMathematicalAnalysis. Dimensions2and3.
13 ANDERSON/FULLER.Ringsand 48 SACHS/WU.GeneralRelativityfor
CategoriesofModules.2nded. Mathematicians.
14 GOLUBITSKY/GUILLEMIN.Stable 49 GRUENBERG/WEIR.LinearGeometry.
MappingsandTheirSingularities. 2nded.
15 BERBERIAN.LecturesinFunctional 50 EDWARDS.Fermat’sLastTheorem.
AnalysisandOperatorTheory. 51 KLINGENBERG.ACourseinDifferential
16 WINTER.TheStructureofFields. Geometry.
17 ROSENBLATT.RandomProcesses.2nded. 52 HARTSHORNE.AlgebraicGeometry.
18 HALMOS.MeasureTheory. 53 MANIN.ACourseinMathematicalLogic.
19 HALMOS.AHilbertSpaceProblemBook. 54 GRAVER/WATKINS.Combinatoricswith
2nded. EmphasisontheTheoryofGraphs.
20 HUSEMOLLER.FibreBundles.3rded. 55 BROWN/PEARCY.Introductionto
21 HUMPHREYS.LinearAlgebraicGroups. OperatorTheoryI:ElementsofFunctional
22 BARNES/MACK.AnAlgebraic Analysis.
IntroductiontoMathematicalLogic. 56 MASSEY.AlgebraicTopology:An
23 GREUB.LinearAlgebra.4thed. Introduction.
24 HOLMES.GeometricFunctionalAnalysis 57 CROWELL/FOX.IntroductiontoKnot
andItsApplications. Theory.
25 HEWITT/STROMBERG.RealandAbstract 58 KOBLITZ.p-adicNumbers,p-adic
Analysis. Analysis,andZeta-Functions.2nded.
26 MANES.AlgebraicTheories. 59 LANG.CyclotomicFields.
27 KELLEY.GeneralTopology. 60 ARNOLD.MathematicalMethodsin
28 ZARISKI/SAMUEL.CommutativeAlgebra. ClassicalMechanics.2nded.
Vol.I. 61 WHITEHEAD.ElementsofHomotopy
29 ZARISKI/SAMUEL.CommutativeAlgebra. Theory.
Vol.II. 62 KARGAPOLOV/MERIZJAKOV.
30 JACOBSON.LecturesinAbstractAlgebra FundamentalsoftheTheoryofGroups.
I.BasicConcepts. 63 BOLLOBAS.GraphTheory.
31 JACOBSON.LecturesinAbstract 64 EDWARDS.FourierSeries.Vol.I.2nded.
AlgebraII.LinearAlgebra. 65 WELLS.DifferentialAnalysisonComplex
32 JACOBSON.LecturesinAbstractAlgebra Manifolds.2nded.
III.TheoryofFieldsandGaloisTheory. 66 WATERHOUSE.IntroductiontoAffine
33 HIRSCH.DifferentialTopology. GroupSchemes.
34 SPITZER.PrinciplesofRandomWalk. 67 SERRE.LocalFields.
2nded. 68 WEIDMANN.LinearOperatorsinHilbert
35 ALEXANDER/WERMER.SeveralComplex Spaces.
VariablesandBanachAlgebras.3rded. 69 LANG.CyclotomicFieldsII.
(continuedafterindex)
M. Ram Murty
Problems in Analytic
Number Theory
Second Edition
(cid:65)(cid:66)(cid:67)
M.RamMurty
DepartmentofMathematics&Statistics
Queen’sUniversity
99UniversityAvenue
KingstonONK7L3N6
Canada
[email protected]
EditorialBoard
S.Axler K.A.Ribet
MathematicsDepartment MathematicsDepartment
SanFranciscoStateUniversity UniversityofCalifornia,Berkeley
SanFrancisco,CA94132 Berkeley,CA94720-3840
USA USA
ISBN978-0-387-72349-5 e-ISBN978-0-387-72350-1
DOI:10.1007/978-0-387-72350-1
LibraryofCongressControlNumber:2007940479
(cid:176)c2008SpringerScience+BusinessMedia,LLC
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,
NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Use
inconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computer
software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
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Likefireinapieceofflint,knowledgeexistsinthemind.
Suggestionisthefrictionwhichbringsitout.
Vivekananda
Preface to the Second Edition
This expanded and corrected second edition has a new chapter on
the important topic of equidistribution. Undoubtedly, one cannot
giveanexhaustivetreatmentofthesubjectinashortchapter.How-
ever,wehopethattheproblemspresentedhereareenticingthatthe
studentwillpursuefurtherandlearnfromothersources.
A problem style presentation of the fundamental topics of ana-
lyticnumbertheoryhasitsvirtues,asIhaveheardfromthosewho
benefitedfromthefirstedition.Meretheoreticalknowledgeinany
fieldisinsufficientforafullappreciationofthesubjectandoneof-
ten needs to grapple with concrete questions in which these ideas
areusedinavitalway.Knowledgeandthevariouslayersof“know-
ing” are difficult to define or describe. However, one learns much
and gains insight only through practice. Making mistakes is an in-
tegral part of learning. Indeed, “it is practice first and knowledge
afterwards.”
Kingston,Ontario,Canada,September2007 M.RamMurty
Acknowledgments for the Second
Edition
I would like to thank several people who have assisted me in
correcting and expanding the first edition. They are Amir Akbary,
Robin Chapman, Keith Conrad, Chantal David, Brandon Fodden,
Sanoli Gun, Wentang Kuo, Yu-Ru Liu, Kumar Murty, Purusottam
RathandMichaelRubinstein.
Kingston,Ontario,Canada,September2007 M.RamMurty
Preface to the First Edition
“In order to become proficient in mathematics, or in any subject,”
writes Andre´ Weil, “the student must realize that most topics in-
volveonlyasmallnumberofbasicideas.”Afterlearningthesebasic
concepts and theorems, the student should “drill in routine exer-
cises, by which the necessary reflexes in handling such concepts
maybeacquired....Therecanbenorealunderstandingofthebasic
conceptsofamathematicaltheorywithoutanabilitytousethemin-
telligently and apply them to specific problems.” Weil’s insightful
observation becomes especially important at the graduate and re-
searchlevel.Itistheviewpointofthisbook.Ourgoalistoacquaint
thestudentwiththemethodsofanalyticnumbertheoryasrapidly
aspossiblethroughexamplesandexercises.
Any landmark theorem opens up a method of attacking other
problems.Unlessthestudentisabletosiftoutfromthemassofthe-
ory 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
generalmethodofDirichletserieswithwhichonecanstudytheas-
ymptotics of sequences. It has also motivated the development of
sieve methods. We focus on this theme in the book. We also touch
upontheemergingSelbergtheory(inChapter8)andp-adicanalytic
numbertheory(inChapter10).
xii Preface
This book is a collection of about five hundred problems in ana-
lyticnumbertheorywiththesingularpurposeoftrainingthebegin-
ninggraduatestudentinsomeofitssignificanttechniques.Assuch,
it is expected that the student has had at least a semester course in
each of real and complex analysis. The problems have been orga-
nizedwiththepurposeofself-instruction.Thosewhoexercisetheir
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 indispens-
ableattheresearchlevel.
The book is ideal for a first course in analytic number theory ei-
ther 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,and8.Anintensegraduatecoursecaneasilycovertheentiretext
in one semester, relegating some of the routine chapters such as
chapters6,7,and10tostudentpresentations.Oronecantakeupa
chapteraweekduringasemestercoursewiththeinstructorfocus-
ingonthemaintheoremsandillustratingthemwithafewworked
examples.
In the course of training students for graduate research, I found
it tedious to keep repeating the cyclic pattern of courses in ana-
lytic and algebraic number theory. This book, along with my other
book“ProblemsinAlgebraicNumberTheory”(writtenjointlywith
J.Esmonde),whichappearsasGraduateTextsinMathematics,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
isa“basictoolkit”forthenumbertheoristandsomeoftheharder
exercisesrevealthesubtle“tricksofthetrade.”
Unless the mind is challenged, it does not perform. The student
is therefore advised to work through the questions with some at-
tention to the time factor. “Work expands to fill the time allotted
toit”andsoifnoupperlimitisassigned,theminddoesnotgetfo-
cused.Thereisnouniversalruleonhowlongoneshouldworkona
problem.However,itisawell-knownfactthatself-discipline,what-
evershapeitmaytake,opensthedoorforinspiration.Ifthemental
muscles are exercised in this fashion, the nuances of the solution
Preface xiii
become clearer and significant. In this way, it is hoped that many,
whodonothaveaccesstoan“externalteacher”willbenefitbythe
approachofthistextandawakentheir“internalteacher.”
Princeton,November1999 M.RamMurty