Trace_formula_in_noncommutative_geometry_and_the_zeros_of_the_Riemann_zeta_function.pdf

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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