Multiplicative Number Theory - I. Classical Theory

571 Pages·2006·4.99 MB·english

by  Hugh L. Montgomery

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MULTIPLICATIVENUMBERTHEORYI:
CLASSICALTHEORY
Primenumbersarethemultiplicativebuildingblocksofnaturalnumbers.Un-
derstanding their overall influence and especially their distribution gives rise
tocentralquestionsinmathematicsandphysics.Inparticulartheirfinerdistri-
butioniscloselyconnectedwiththeRiemannhypothesis,themostimportant
unsolvedprobleminthemathematicalworld.Assumingonlysubjectscovered
inastandarddegreeinmathematics,theauthorscomprehensivelycoverallthe
topicsmetinfirstcoursesonmultiplicativenumbertheoryandthedistribution
ofprimenumbers.Theybringtheirextensiveanddistinguishedresearchexper-
tisetobearinpreparingthestudentforintelligentreadingofthemoreadvanced
researchliterature.Thetext,whichisbasedoncoursestaughtsuccessfullyover
manyyearsatMichigan,ImperialCollegeandPennsylvaniaState,isenriched
bycomprehensivehistoricalnotesandreferencesaswellasover500exercises.
HughMontgomeryisaProfessorofMathematicsattheUniversityofMichigan.
Robert Vaughan is a Professor of Mathematics at Pennsylvannia State
University.
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74 R.M.DudleyRealanalysisandprobability
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Multiplicative Number Theory
I. Classical Theory
HUGH L. MONTGOMERY
UniversityofMichigan,AnnArbor
ROBERT C. VAUGHAN
PennsylvaniaStateUniversity,UniversityPark
cambridge university press
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Cambridge University Press
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© Cambridge University Press 2006
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof
relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace
withoutthewrittenpermissionofCambridgeUniversityPress.
Firstpublishedinprintformat 2006
isbn-13 978-0-511-25645-5eBook(EBL)
isbn-10 0-511-25645-0 eBook(EBL)
isbn-13 978-0-521-84903-6hardback
isbn-10 0-521-84903-9 hardback
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Dedicatedtoourteachers:
P.T.Bateman
J.H.H.Chalk
H.Davenport
T.Estermann
H.Halberstam
A.E.Ingham
Taleta¨rta¨nkandetsbo¨rjanochslut.
Medtankenfo¨ddestalet.
Uto¨fvertaletn˚artankenicke.
Numbersarethebeginningandendofthinking.
Withthoughtswerenumbersborn.
Beyondnumbersthoughtdoesnotreach.
Magnus Gustaf Mittag-Leffler, 1903
Contents
Preface pagexi
Listofnotation xiii
1 Dirichletseries:I 1
1.1 Generatingfunctionsandasymptotics 1
1.2 AnalyticpropertiesofDirichletseries 11
1.3 Eulerproductsandthezetafunction 19
1.4 Notes 31
1.5 References 33
2 Theelementarytheoryofarithmeticfunctions 35
2.1 Meanvalues 35
2.2 TheprimenumberestimatesofChebyshevandofMertens 46
2.3 Applicationstoarithmeticfunctions 54
2.4Thedistributionof (cid:1)(n)−ω(n)65
2.5 Notes 68
2.6 References 71
3 Principlesandfirstexamplesofsievemethods 76
3.1 Initiation 76
3.2TheSelberglambda-squaredmethod82
3.3 Siftinganarithmeticprogression 89
3.4 Twinprimes 91
3.5 Notes 101
3.6 References 104
4 Primesinarithmeticprogressions:I 108
4.1 Additivecharacters 108
4.2 Dirichletcharacters 115
4.3 Dirichlet L-functions 120
vii
viii Contents
4.4 Notes 133
4.5 References 134
5 Dirichletseries:II 137
5.1 TheinverseMellintransform 137
5.2 Summability 147
5.3 Notes 162
5.4 References 164
6 ThePrimeNumberTheorem 168
6.1 Azero-freeregion 168
6.2 ThePrimeNumberTheorem 179
6.3 Notes 192
6.4 References 195
7 ApplicationsofthePrimeNumberTheorem 199
7.1 Numberscomposedofsmallprimes 199
7.2 Numberscomposedoflargeprimes 215
7.3 Primesinshortintervals 220
7.4 Numberscomposedofaprescribednumberofprimes 228
7.5 Notes 239
7.6 References 241
8 FurtherdiscussionofthePrimeNumberTheorem 244
8.1 RelationsequivalenttothePrimeNumberTheorem 244
8.2 AnelementaryproofofthePrimeNumberTheorem 250
8.3 TheWiener–IkeharaTauberiantheorem 259
8.4 Beurling’sgeneralizedprimenumbers 266
8.5 Notes 276
8.6 References 279
9 PrimitivecharactersandGausssums 282
9.1 Primitivecharacters 282
9.2 Gausssums 286
9.3 Quadraticcharacters 295
9.4 Incompletecharactersums 306
9.5 Notes 321
9.6 References 323
10 Analyticpropertiesofthezetafunctionand L-functions 326
10.1 Functionalequationsandanalyticcontinuation 326
10.2 Productsandsumsoverzeros 345
10.3 Notes 356
10.4 References 356