Equivalents of the Riemann Hypothesis I: Arithmetic Equivalents

348 Pages·2017·9.38 MB·english

by  Kevin Broughan

EQUIVALENTS OF THE RIEMANN HYPOTHESIS
VolumeOne:ArithmeticEquivalents
TheRiemannhypothesis(RH)isperhapsthemostimportantoutstandingproblem
inmathematics.Thistwo-volumetextpresentsthemainknownequivalentstoRH
usinganalyticandcomputationalmethods.Thebooksaregentleonthereaderwith
definitionsrepeated,proofssplitintologicalsections,andgraphicaldescriptionsof
therelationsbetweendifferentresults.Theyalsoincludeextensivetables,
supplementarycomputationaltools,andopenproblemssuitableforresearch.
Accompanyingsoftwareisfreetodownload.
Thesebookswillinterestmathematicianswhowishtoupdatetheirknowledge,
graduateandseniorundergraduatestudentsseekingaccessibleresearchproblemsin
numbertheory,andotherswhowanttoexploreandextendresultscomputationally.
Eachvolumecanbereadindependently.
Volume1presentsclassicalandmodernarithmeticequivalentstoRH,withsome
analyticmethods.Volume2coversequivalenceswithastronganalyticorientation,
supportedbyanextensivesetofappendicescontainingfullydevelopedproofs.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercises
attheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
Encyclopedia of Mathematics and Its Applications
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
119 M.DezaandM.DutourSikiric´GeometryofChemicalGraphs
120 T.NishiuraAbsoluteMeasurableSpaces
121 M.PrestPurity,SpectraandLocalisation
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopological
GraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,Computer
Science,andEngineering
135 V.Berthe´andM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Krista´ly,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,
Geometry,andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland
DifferentialEquations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethod
forMaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunn–MinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.CabreraGarc´ıaandA´.Rodr´ıguezPalaciosNon-AssociativeNormedAlgebrasI:The
Vidav–PalmerandGelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory
157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.MoraSolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond
159 V.Berthe´andM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics
160 B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonic
Analysis
161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractional
Problems
163 S.WagonTheBanach–TarskiParadox(SecondEdition)
164 K.BroughanEquivalentsoftheRiemannHypothesisI:ArithmeticEquivalents
165 K.BroughanEquivalentsoftheRiemannHypothesisII:AnalyticEquivalents
166 M.BaakeandU.GrimmAperiodicOrderII:RepresentationTheoryandtheZelmanovApproach
Encyclopedia of Mathematics and Its Applications
Equivalents of the Riemann Hypothesis
Volume One: Arithmetic Equivalents
KEVIN BROUGHAN
UniversityofWaikato,NewZealand
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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Informationonthistitle:www.cambridge.org/9781107197046
DOI:10.1017/9781108178228
(cid:2)c KevinBroughan2017
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2017
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Broughan,KevinA.(KevinAlfred),1943–author.
Title:EquivalentsoftheRiemannhypothesis/KevinBroughan,UniversityofWaikato,
NewZealand.
Description:Cambridge:CambridgeUniversityPress,2017–|
Series:Encyclopediaofmathematicsanditsapplications;164|
Includesbibliographicalreferencesandindex.Contents:volume1.Arithmeticequivalents
Identifiers:LCCN2017034308|ISBN9781107197046(hardback:alk.paper:v.1)
Subjects:LCSH:Riemannhypothesis.Classification:LCCQA246.B7452017|
DDC512.7/3–dc23
LCrecordavailableathttps://lccn.loc.gov/2017034308
ISBN–2VolumeSet978-1-108-29078-4Hardback
ISBN–Volume1978-1-107-19704-6Hardback
ISBN–Volume2978-1-107-19712-1Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof
URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
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DedicatedtoJackie,JudeandBeck
RH isaprecisestatement,andinone sensewhat itmeansisclear,butwhat
it is connected with, what it implies, where it comes from, can be very un-
obvious.
MartinHuxley
Contents for Volume One
ContentsforVolumeTwo pagex
ListofIllustrations xiv
ListofTables xvi
PrefaceforVolumeOne xvii
ListofAcknowledgements xxi
1 Introduction 1
1.1 ChapterSummary 1
1.2 EarlyHistory 1
1.3 VolumeOneSummary 8
1.4 Notation 12
1.5 BackgroundReading 13
1.6 UnsolvedProblems 14
2 TheRiemannZetaFunction 15
2.1 Introduction 15
2.2 BasicProperties 16
2.3 Zero-FreeRegions 21
2.4 Landau’sZero-FreeRegion 25
2.5 Zero-FreeRegionsSummary 29
2.6 TheProductOverZetaZeros 30
2.7 UnsolvedProblems 39
3 Estimates 40
3.1 Introduction 40
3.2 ConstructingTablesofBoundsforψ(x) 41
3.3 ExactVerificationUsingComputation 51
3.4 Estimatesforθ(x) 54
3.5 MoreEstimates 65
3.6 UnsolvedProblems 67
vii
viii ContentsforVolumeOne
4 ClassicalEquivalences 68
4.1 Introduction 68
4.2 ThePrimeNumberTheoremandItsRHEquivalences 69
4.3 OscillationTheorems 81
4.4 ErrorsinArithmeticSums 88
4.5 UnsolvedProblems 93
5 Euler’sTotientFunction 94
5.1 Introduction 94
5.2 EstimatesforEuler’sFunctionϕ(n) 98
5.3 PreliminaryResultsWithRHTrue 110
5.4 FurtherResultsWithRHTrue 123
5.5 PreliminaryResultsWithRHFalse 130
5.6 Nicolas’FirstTheorem 135
5.7 Nicolas’SecondTheorem 137
5.8 UnsolvedProblems 142
6 AVarietyofAbundantNumbers 144
6.1 Introduction 144
6.2 SuperabundantNumbers 147
6.3 ColossallyAbundantNumbers 153
6.4 Estimatesforx ((cid:5)) 161
2
6.5 UnsolvedProblems 163
7 Robin’sTheorem 165
7.1 Introduction 165
7.2 Ramanujan’sTheoremAssumingRH 169
7.3 PreliminaryLemmasWithRHTrue 174
(cid:2)
7.4 Bounding p≤x(1−p−2)FromAboveWithRHTrue 180
7.5 BoundingloglogNFromBelowWithRHTrue 184
7.6 ProofofRobin’sTheoremWithRHTrue 186
7.7 AnUnconditionalBoundforσ(n)/n 188
7.8 BoundingloglogNFromAboveWithoutRH 190
7.9 ALowerBoundforσ(n)/nWithRHFalse 191
7.10 Lagarias’FormulationofRobin’sCriterion 193
7.11 UnconditionalResultsforLagarias’Formulation 196
7.12 UnitaryDivisorSums 197
7.13 UnsolvedProblems 198
8 NumbersThatDoNotSatisfyRobin’sInequality 200
8.1 Introduction 200
8.2 Hardy–RamanujanNumbers 202
8.3 IntegersNotDivisiblebytheFifthPowerofAnyPrime 208