Equivalents of the Riemann Hypothesis II Analytic Equivalents

512 Pages·2017·7.93 MB·english

by  Kevin Broughan

EQUIVALENTS OF THE RIEMANN HYPOTHESIS
VolumeTwo:AnalyticEquivalents
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 Two: Analytic Equivalents
KEVIN BROUGHAN
UniversityofWaikato,NewZealand
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Informationonthistitle:www.cambridge.org/9781107197121
DOI:10.1017/9781108178266
(cid:2)c KevinBroughan2017
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
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Firstpublished2017
PrintedintheUnitedKingdombyClays,StIvesplc
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloguinginPublicationData
Names:Broughan,KevinA.(KevinAlfred),1943–author.
Title:EquivalentsoftheRiemannhypothesis/KevinBroughan,
UniversityofWaikato,NewZealand.
Description:Cambridge:CambridgeUniversityPress,2017–|
Series:Encyclopediaofmathematicsanditsapplications;165|
Includesbibliographicalreferencesandindex.Contents:volume2.AnalyticEquivalents
Identifiers:LCCN2017034308|ISBN9781107197121(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
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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 Two
ContentsforVolumeOne pagexi
ListofIllustrations xiv
ListofTables xvi
PrefaceforVolumeTwo xvii
ListofAcknowledgements xx
1 Introduction 1
1.1 WhyThisStudy? 1
1.2 SummaryofVolumeTwo 2
1.3 HowtoReadThisBook 7
2 SeriesEquivalents 8
2.1 Introduction 8
2.2 TheRieszFunction 10
2.3 AdditionalPropertiesoftheRieszFunction 14
2.4 TheSeriesofHardyandLittlewood 15
2.5 AGeneralTheoremforaClassofEntireFunctions 16
2.6 FurtherWork 22
3 BanachandHilbertSpaceMethods 23
3.1 Introduction 23
3.2 PreliminaryDefinitionsandResults 25
3.3 Beurling’sTheorem 29
3.4 RecentDevelopments 35
4 TheRiemannXiFunction 37
4.1 Introduction 37
4.2 PreliminaryResults 40
4.3 Monotonicityof|ξ(s)| 49
vii
viii ContentsforVolumeTwo
4.4 PositiveEvenDerivatives 51
4.5 Li’sEquivalence 54
4.6 MoreRecentResults 59
5 TheDeBruijn–NewmanConstant 62
5.1 Introduction 62
5.2 PreliminaryDefinitionsandResults 66
5.3 ARegionforΞλ(z)WithOnlyRealZeros 69
5.4 TheExistenceofΛ 77
5.5 ImprovedLowerBoundsforΛ 77
5.5.1 Lehmer’sPhenomenon 78
5.5.2 TheDifferentialEquationSatisfiedbyH(t,z) 81
5.5.3 FindingaLowerBoundforΛ UsingLehmerPairs 87
C
5.6 FurtherWork 92
6 OrthogonalPolynomials 93
6.1 Introduction 93
6.2 Definitions 94
6.3 OrthogonalPolynomialProperties 96
6.4 Moments 99
6.5 Quasi-AnalyticFunctions 104
6.6 Carleman’sInequality 106
6.7 RiemannZetaFunctionApplication 113
6.8 RecentWork 116
7 CyclotomicPolynomials 117
7.1 Introduction 117
7.2 Definitions 118
7.3 PreliminaryResults 119
7.4 RiemannHypothesisEquivalences 124
7.5 FurtherWork 126
8 IntegralEquations 127
8.1 Introduction 127
8.2 PreliminaryResults 129
8.3 TheMethodofSekatskii,BeltraminelliandMerlini 133
8.4 Salem’sEquation 139
8.5 Levinson’sEquivalence 142
9 Weil’sExplicitFormula,InequalityandConjectures 150
9.1 Introduction 150
9.2 Definitions 152
9.3 PreliminaryResults 152
9.4 Weil’sExplicitFormula 154