Table Of Content“FM” — 2018/10/29 — 18:54 — page i — #1
MATHEMATICAL CONSTANTS II
Famousmathematicalconstantsincludetheratioofcircularcircumferenceto
diameter,(cid:25)=3:14:::;andthenaturallogarithmbase,e=2:718::::Students
andprofessionalscanoftennameafewothers,buttherearemanymoreburied
intheliteratureawaitingdiscovery.
Howdosuchconstantsarise,andwhyaretheyimportant?Heretheauthor
renewsthesearchhebeganinhisbookMathematicalConstants,adding
another133essaysthatbroadenthelandscape.Topicsincludetheminimalityof
soapfilmsurfaces,primenumbers,ellipticcurvesandmodularforms,
Poisson–Voronoitessellations,randomtriangles,Brownianmotion,uncertainty
inequalities,Prandtl–Blasiusflow(fromfluiddynamics),Lyapunovexponents,
knotsandtangles,continuedfractions,Galton–Watsontrees,electrical
capacitance(frompotentialtheory),Zermelo’snavigationproblem,andthe
optimalcontrolofapendulum.Unsolvedproblemsappearvirtually
everywhereaswell.Thisvolumecontinuesanoutstandingscholarlyattemptto
bringtogetherallsignificantmathematicalconstantsinoneplace.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplication
inmathematicsormathematicalscienceandforwhichadetaileddevelopment
oftheabstracttheoryislessimportantthanathoroughandconcrete
explorationoftheimplicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedas
exercisesattheendsofchapters.Fortechnicalities,readersarereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
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ENCYCLOPEDIAOFMATHEMATICSANDITSAPPLICATIONS
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
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.BerthéandM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Kristály,V.D.RădulescuandC.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.NovikovIntroductiontotheNetworkApproximation
MethodforMaterialsModeling
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.CabreraGarcíaandÁ.RodríguezPalaciosNon-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.BerthéandM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics
160 B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusand
HarmonicAnalysis
161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.D.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractional
Problems
163 S.WagonTheBanach–TarskiParadox(SecondEdition)
164 K.BroughanEquivalentsoftheRiemannHypothesisI:ArithmeticEquivalents
165 K.BroughanEquivalentsoftheRiemannHypothesisII:AnalyticEquivalents
166 M.BaakeandU.Grimm(eds.)AperiodicOrderII:CrystallographyandAlmostPeriodicity
167 M.CabreraGarcíaandÁ.RodríguezPalaciosNon-AssociativeNormedAlgebrasII:
RepresentationTheoryandtheZel’manovApproach
168 A.Yu.Khrennikov,S.V.KozyrevandW.A.Zún~iga-GalindoUltrametricPseudodifferential
EquationsandApplications
“FM” — 2018/10/29 — 18:54 — page iii — #3
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Mathematical Constants II
STEVEN R. FINCH
MassachusettsInstituteofTechnology
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UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India
79AnsonRoad,#06-04/06,Singapore079906
CambridgeUniversityPressispartoftheUniversityofCambridge.
ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof
education,learning,andresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108470599
DOI:10.1017/9781316997741
©StevenR.Finch2019
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2019
PrintedintheUnitedKingdombyTJInternationalLtd.PadstowCornwall
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
ISBN978-1-108-47059-9Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
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Contents
Preface pageix
Notation xi
1 NumberTheoryandCombinatorics 1
1.1 Bipartite,k-Colorableandk-ColoredGraphs 1
1.2 TransitiveRelations,TopologiesandPartialOrders 7
1.3 Series-ParallelNetworks 12
1.4 TwoAsymptoticSeries 21
1.5 MultiplesandDivisors 32
1.6 DiscrepancyandUniformity 40
1.7 UnitarismandInfinitarism 49
1.8 Erdős’MinimumOverlapProblem 56
1.9 PlanarGraphGrowthConstants 58
1.10 TauberianConstants 67
1.11 IntegerPartitions 72
1.12 ClassNumberTheory 78
1.13 QuadraticDirichletL-Series 97
1.14 EllipticCurvesoverQ 112
1.15 ModularFormsonSL (Z) 132
2
1.16 Chebyshev’sBias 147
1.17 Pattern-AvoidingPermutations 149
1.18 CyclicGroupOrders 154
1.19 DedekindEtaProducts 157
1.20 SeriesinvolvingArithmetricFunctions 160
1.21 RiemannZetaMoments 176
1.22 CentralBinomialCoefficients 182
1.23 FractionalPartsofBernoulliNumbers 189
1.24 ProductsofConsecutive-IntegerRatios 192
1.25 PrimeNumberTheorem 195
1.26 Mertens’Formula 202
1.27 CyclotomicPolynomials 206
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vi Contents
1.28 Minkowski–AlkauskasConstant 207
1.29 Two-ColoringsofPositiveIntegers 209
1.30 SignumEquationsandExtremalCoefficients 213
1.31 MonoidsofNaturalNumbers 217
1.32 PrimitiveCuspForm 223
1.33 CubicandQuarticCharacters 232
1.34 DistributionofErrorTerms 240
1.35 Cilleruelo’sLCMConstants 242
1.36 AmicablePairsandAliquotSequences 244
1.37 FermatNumbersandElitePrimes 247
1.38 AverageLeastNonresidues 249
1.39 ApollonianCircleswithIntegerCurvatures 253
1.40 Molteni’sCompositionConstant 258
1.41 BooleanDecisionFunctions 260
1.42 MapAsymptoticsConstant 263
1.43 Injections,SurjectionsandMore 266
2 InequalitiesandApproximation 272
2.1 Hardy–LittlewoodMaximalInequalities 272
2.2 BesselFunctionZeroes 275
2.3 Nash’sInequality 284
2.4 UncertaintyInequalities 291
2.5 AiryFunctionZeroes 295
2.6 ProjectionsofMinimalNorm 299
2.7 Bohr’sInequality 303
2.8 ModuliofContinuity 306
2.9 Quinn–Rand–StrogatzConstant 311
2.10 Tsirelson’sConstant 314
2.11 MathieuEigenvalues 321
2.12 Thomas–FermiModel 324
2.13 Prandtl–BlasiusFlow 329
2.14 Lane–Ritter–EmdenConstants 338
2.15 RadiativeTransferEquations 344
2.16 Carleman’sInequality 352
2.17 Golay–LittlewoodProblem 354
2.18 OnlineMatchingCoins 358
2.19 ToothpicksandLiveCells 363
2.20 VirialCoefficients 365
2.21 StrongTriangleInequality 373
3 RealandComplexAnalysis 377
3.1 RadiiinGeometricFunctionTheory 377
3.2 NumericalRadiiofLinearOperators 387
3.3 CoefficientEstimatesforUnivalentFunctions 390
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Contents vii
3.4 PlanarHarmonicMappings 399
3.5 ConstantofInterpolation 403
3.6 DirichletIntegral 406
3.7 BrachistochroneProblem 407
3.8 UnconditionalBasisConstants 410
3.9 PowerSerieswithRestrictedCoefficients 414
3.10 HankelandToeplitzDeterminants 415
3.11 Goˇldberg’sZero-OneConstants 419
3.12 ElectricalCapacitance 423
3.13 Aissen’sConvexSetFunction 428
3.14 ConditionNumbersofMatrices 430
3.15 Goddard’sRocketProblem 432
3.16 Swing-UpControlofaPendulum 437
3.17 Zermelo’sNavigationProblem 444
4 ProbabilityandStochasticProcesses 455
4.1 Hammersley’sPathProcess 455
4.2 MomentsofSums 462
4.3 Ornstein–UhlenbeckProcess 468
4.4 ZeroCrossings 479
4.5 VariantsofBrownianMotion 485
4.6 ShapesofBinaryTrees 494
4.7 ExpectedLifetimesandInradii 504
4.8 SubcriticalGalton–WatsonTrees 511
4.9 ContinuedFractionTransformation 519
4.10 ContinuedFractionTransformation.II 527
4.11 ContinuedFractionTransformation.III 533
4.12 ContinuedFractionTransformation.IV 541
4.13 LyapunovExponents 552
4.14 LyapunovExponents.II 558
4.15 LyapunovExponents.III 564
4.16 LyapunovExponents.IV 568
4.17 StarsandWatermelons 577
4.18 ProphetInequalities 580
4.19 ExcursionDurations 582
4.20 Gambler’sRuin 584
4.21 Self-Convolutions 589
4.22 Newcomb–BenfordLaw 592
4.23 ElectingaLeader 594
4.24 SubstitutionDynamics 599
4.25 Biham–Middleton–LevineTraffic 604
4.26 ContactProcesses 605
4.27 InterpolatingbetweenMaxandSum 612
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viii Contents
4.28 MixingTimeofMarkovChains 616
4.29 CorrelatedProducts 619
5 GeometryandTopology 623
5.1 Knots,LinksandTangles 623
5.2 ConvexLatticePolygons 632
5.3 VolumesofHyperbolic3-Manifolds 636
5.4 Poisson–VoronoiTessellations 643
5.5 OptimalEscapePaths 652
5.6 Minkowski–SiegelMassConstants 654
5.7 SlicingProblem 660
5.8 ConstantofTheodorus 662
5.9 Nearest-NeighborGraphs 664
5.10 RandomTriangles 668
5.11 RandomTriangles.II 682
5.12 RandomTriangles.III 689
5.13 RandomTriangles.IV 695
5.14 RandomTriangles.V 700
5.15 RandomTriangles.VI 713
5.16 CollidingDiceProbabilities 718
5.17 Gergonne–SchwarzSurface 721
5.18 PartitioningProblem 731
5.19 SoapFilmExperiments 740
5.20 InflatinganInelasticMembrane 747
5.21 EnumerativeGeometry 750
5.22 Distance-AvoidingSetsinthePlane 754
5.23 FraenkelAsymmetry 757
Index 765
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Preface
OnereviewerforthefirstvolumeofMathematicalConstantsdescribedthebook
as “excellent bedtime reading” [1]. My aim here is similar to before: to gather
far-flungideasinoneplace,focusingonhighlyconcrete,eminentlycomputable
results.Theseessaysrecountstoriesthatarebothsuccessful(withdepthofunder-
standing) and tangible (in terms of numerical precision). Much mathematical
researchthesedaysisnecessarilyabstractandqualitative,duetotheenormous
difficultyoftheissuesunderconsideration.HereIdirectthespotlighttothose
rare cases when quantitative exactness still appears to be pertinent. My words
fromfifteenyearsago(concerningpurposeandscope)applyaswellnowasthen.
Asampleproblemservestoillustratemyendeavor.Whilediscussingthemass
M of all nonisomorphic Type I inner product modules of rank n, Milnor &
n
Husemoller [2] gave a plot of M on a logarithmic scale for 1(cid:20)n(cid:20)30. They
n
remarked that M is asymptotic to C(cid:1)F(n) as n!1, describing the function
n
F(n) exactly, but reporting only that “the constant C is approximately 0:705”.
Unravelingthisenigma–whatistheprecisenatureofC?–iscaptivatingtome.
Understandablythisquestionwasincidentaltothepurposesof[2];itis,however,
centralheretome[3].Theanswerinvolvesaquantity[4]discoveredin1860,as
wellassomethingelse.
ThisvolumeisdedicatedtothememoryofPhilippeFlajolet,afearlessleader
andinspiringmentor.Itisalsoatributetomyparents,CharlesRichardFinch
andShirleyPeeryFinch,andtomysiblings,ValerieJeanBridge,GregoryCharles
FinchandWilliamRobertFinch,withloveandgratitude.IacknowledgeaBook
Fellowship from the Clay Mathematics Institute in 2004–2005, long before the
magnitudeofmypresenttaskbecameclear.
“Open the book at random”, the aforementioned reviewer wrote, evoking a
fewconstantsfrommanyacrossthecanvas,seedlingsdrawnfromavastforest.
Read,learn,wander,reflect,...“andsoonintothenight”.
