A Strategy for Proving Riemann Hypothesis

arXiv:math/0111262v5[math.GM]25Jan2002AStrategyforProvingRiemannHypothesisM.Pitk¨anen11DepartmentofPhysicalSciences,HighEnergyPhysicsDivision,PL64,FIN-00014,UniversityofHelsinki,[e-mail].fi,fi/∼matpitka/.Recentaddress:Kadermonkatu16,10900,Hanko,Finland.Abstract.AstrategyforprovingRiemannhypothesisissuggested.ThevanishingoftheRiemanZetareducestoanorthogonalityconditionfortheeigenfunctionsofanon-HermitianoperatorD+havingthezerosofRiemannZetaasitseigenvalues.TheconstructionofD+isinspiredbytheconvictionthatRiemannZetaisassociatedwithaphysicalsystemallowingconformaltransformationsasitssymmetries.TheeigenfunctionsofD+areanalogoustothesocalledcoherentstatesandingeneralnotorthogonaltoeachother.Thestatesorthogonaltoavacuumstate(whichhasanegativenormsquared)correspondtothezerosoftheRiemannZeta.TheinducedmetricinthespaceVofstateswhichcorrespondtothezerosoftheRiemannZetaatthecriticallineRe[s]=1/2ishermitianandbothhermiticityandpositivedefinitenesspropertiesimplyRiemannhypothesis.ConformalinvarianceinthesenseofgaugeinvarianceallowsonlythestatesbelongingtoV.RiemannhypothesisfollowsalsofromarestrictedformofadynamicalconformalinvarianceinVandonecanreducetheprooftoastandardanalyticargumentusedinLiegrouptheory.11IntroductionTheRiemannhypothesis[6,7]statesthatthenon-trivialzeros(asopposedtozerosats=−2n,n≥1integer)ofRiemannZetafunctionobtainedbyanalyticallycontinuingthefunctionζ(s)=∞Xn=11ns(1)fromtheregionRe[s]1totheentirecomplexplane,lieonthelineRe[s]=1/2.HilbertandPolyaconjecturedalongtimeagothatthenon-trivialzeroesofRiemannZetafunctioncouldhavespectralinterpretationintermsoftheeigenvaluesofasuitableself-adjointdifferentialoperatorHsuchthattheeigenvaluesofthisoperatorcorrespondtotheimaginarypartsofthenontrivialzerosz=x+iyofζ.Onecanhoweverconsideravariantofthishypothesisstatingthattheeigenvaluespectrumofanon-hermitianoperatorD+containsthenon-trivialzerosofζ.TheeigenstatesinquestionareeigenstatesofanannihilationoperatortypeoperatorD+andanalogoustothesocalledcoherentstatesencounteredinquantumphysics[4].Inparticular,theeigenfunctionsareingeneralnon-orthogonalandthisisaquintessentialelementofthetheproposedstrategyofproof.Inthefollowinganexplicitoperatorhavingasitseigenvaluesthenon-trivialzerosofζisconstructed.a)Theconstructionreliescruciallyontheinterpretationofthevanishingofζasanorthogo-nalityconditioninahermitianmetricwhichisisapriorimoregeneralthanHilbertspaceinnerproduct.b)Secondbasicelementisthescalinginvariancemotivatedbythebeliefthatζisassociatedwithaphysicalsystemwhichhassuperconformaltransformations[3]asitssymmetries.Thecoreelementsoftheconstructionarefollowing.a)AllcomplexnumbersarecandidatesfortheeigenvaluesofD+(formalhermitianconjugateofD)andgenuineeigenvaluesareselectedbytherequirementthattheconditionD†=D+holdstrueinthesetofthegenuineeigenfunctions.Thisconditionisequivalentwiththehermiticityofthemetricdefinedbyafunctionproportionaltoζ.b)Theeigenvaluesturnouttoconsistofz=0andthenon-trivialzerosofζandonlytheeigenfunctionscorrespondingtothezeroswithRe[s]=1/2defineasubspacepossessingahermitianmetric.Thevanishingofζtellsthatthe’physical’positivenormeigenfunctions(ingeneralnotorthogonaltoeachother),areorthogonaltothe’unphysical’negativenormeigenfunctionassociatedwiththeeigenvaluez=0.TheproofoftheRiemannhypothesisbyreductioadabsurdumresultsifoneassumesthatthespaceVspannedbythestatescorrespondingtothezerosofζinsidethecriticalstriphasahermitianinducedmetric.RiemannhypothesisfollowsalsofromtherequirementthattheinducedmetricinthespacessubspacesVsofVspannedbythestatesΨsandΨ1−sdoesnotpossessnegativeeigenvalues:thisconditionisequivalentwiththepositivedefinitenessofthemetricinV.ConformalinvarianceinthesenseofgaugeinvarianceallowsonlythestatesbelongingtoV.RiemannhypothesisfollowsalsofromarestrictedformofadynamicalconformalinvarianceinV.ThisallowsthereductionoftheprooftoastandardanalyticargumentusedinLie-grouptheory.22ModifiedformoftheHilbert-PolyaconjectureOnecanmodifytheHilbert-PolyaconjecturebyassumingscalinginvarianceandgivingupthehermiticityoftheHilbert-Polyaoperator.Thismeansintroductionofthenon-hermitianoper-atorsD+andDwhicharehermitianconjugatesofeachothersuchthatD+hasthenontrivialzerosofζasitscomplexeigenvaluesD+Ψ=zΨ.(2)Thecounterpartsofthesocalledcoherentstates[4]areinquestionandtheeigenfunctionsofD+arenotexpectedtobeorthogonalingeneral.ThefollowingconstructionisbasedontheideathatD+alsoallowstheeigenvaluez=0andthatthevanishingofζatzexpressestheorthogonalityofthestateswitheigenvaluez=x+iy6=0andthestatewitheigenvaluez=0whichturnsouttohaveanegativenorm.ThetrialD=L0+V,D+=−L0+VL0=tddt,V=dlog(F)d(log(t))=tdFdt1F(3)ismotivatedbytherequirementofinvariancewithrespecttoscalingst→λtandF→λF.Therangeofvariationforthevariabletconsistsofnon-negativerealnumberst≥0.Thescalinginvarianceimplyingconformalinvariance(VirasorogeneratorL0representsscalingwhichplaysafundamentalroleinthesuperconformaltheories[3])ismotivatedbythebeliefthatζcodesforthephysicsofaquantumcriticalsystemhaving,notonlysupersymmetri