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时间:2019-03-01
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1、TraceformulainnoncommutativeGeometryandthezerosoftheRiemannzetafunctionAlainCONNESAbstractWegiveaspectralinterpretationofthecriticalzerosoftheRie-mannzetafunctionasanabsorptionspectrum,whileeventualnoncriticalze-rosappearasresonances.Wegiveageometricinterpretationoftheexplicitformulasofnumb
2、ertheoryasatraceformulaonthenoncommutativespaceofAdeleclasses.ThisreducestheRiemannhypothesistothevalidityofthetraceformulaandeliminatestheparameterδofourpreviousapproach.TableofcontentsIntroduction.IQuantumchaosandthehypotheticalRiemannflow.IIAlgebraicGeometryandglobalfieldsofnonzerocharacte
3、ristic.arXiv:math/9811068v1[math.NT]10Nov1998IIISpectralinterpretationofcriticalzeros.IVThedistributiontraceformulaforflowsonmanifolds.VTheaction(λ,x)→λxofK∗onalocalfieldK.VITheglobalcase,andtheformaltracecomputation.VIIProofofthetraceformulaintheS-localcase.VIIIThetraceformulaintheglobalcase
4、,andeliminationofδ.AppendixI,Proofoftheorem1.AppendixII,Explicitformulas.AppendixIII,Distributiontraceformulas.1IntroductionWeshallgiveinthispaperaspectralinterpretationofthezerosoftheRiemannzetafunctionandageometricframeworktowhichonecantransposetheideasofalgebraicgeometryinvolvingtheactio
5、noftheFrobeniusandtheLefchetzformula.Thespectralinterpretationofthezerosofzetawillbeasanabsorptionspectrum,i.e.asmissingspectrallines.Allzeroswillplayaroleinthespectralsideofthetraceformula,butwhilethecriticalzeroswillappearper-se,thenoncriticaloneswillappearasresonancesandenterinthetracefo
6、rmulathroughtheirharmonicpotentialwithrespecttothecriticalline.Thusthespectralsideisentirelycanonical,andbyprovingpositivityoftheWeildistribution,weshallshowthatitsequalitywiththegeometricside,i.e.theglobaltraceformula,isequivalenttotheRiemannHypothesisforallL-functionswithGr¨ossencharakter
7、.WeshallmodelourdiscussionontheSelbergtraceformula,butitdiffersfromthelatterinseveralimportantrespects.Weshallfirstexplaininparticularwhyacrucialnegativesignintheanalysisofthestatisticalfluctuationsofthezerosofzetaindicatesthatthespectralinterpretationshoul
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