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1、数理解析研究所講究録1590巻2008年126-145ThreeNotesonConnectionsbetweentheRiemannZetaFunctionandProbabilityTheory,inparticular:RandomMatrixTheory.M.Yor(1)$(2)$March12,2008(1)Laboratoirede$Probabilit6s$et$Mod6lesAl6atoires$,$Universit68$ParisVIetVII,4PlaceJussieu-Case188,F-75252ParisCedex05E–mail:deaprob
2、aGproba.juss$i$eu.$fr$(2)Institut$Univer8itaire$deFranceA-RandomMatricesandtheRiemannZetafunction:theKeating-Snaithphilosophy$B$-AfurthernoteonSelberg’sintegrals,inspiredbyN.Snaith’sresultsaboutthedistributionofsomecharacteristicpolynomials$C$-OnthelogarithmoftheRiemannZetafunction:fromSel
3、berg’scentrallimittheoremtototaldisorder$************$SomepertinentcommentsabouteachoftheNotes$A,$$B,$$C$havebeenmadebyP.Bourgade;theyarepresentedjustafterC.127A-RandomMatricesandtheRiemannZetafunction:theKeating-SnaithphilosophyM.Yor(1)$(2)$March32008(1)Laboratoirede$Probabilit68$etModele
4、s$Al6atoires$,Universit&ParisVIetVII,4PlaceJussieu-Case188,F-75252$Pari8$Cedex05E–mail:$dea_{P^{robaQP}^{roba.j}}$us$si$eu.$fr$(2)InstitutUniversitairedeFtanceAbstractTheextremelypreciseconjectureofKeating-Snaithabouttheasymp-toticsofthemomentsoftheRiemannZetafunctiononthecriticalline,asth
5、eheight$T$tends$to+infty$ispresented,togetherwithsomestrikingsimilaritiesbetweentheRiemannZetaasymptotics,as$Tarrowinfty$,andasymptoticsaboutthegenericmatrix$A_{N}$ontheunitarygroup$U_{N}$,as$Narrowinfty$.ExplicitMellin-FouriercomputationsdonebyKeating-Snaithabout$(A_{N})$areinterpreted
6、probabilistically.Furtherheunisticsforthe$(KS)$conjecturearealsodiscussed.1TheKeating-Snaithconjecture(1.1)TheimportanceoftheRiemannHypothesis:Allnon-trivialzerosoftheZetafunction$(RH)$${rmRe}(s)=underline{1}$$(zeta(s);sinmathbb{C}backslash{1})$lieonthecriticalline:justifiestheinten
7、sivestudieswhichkeepbeingdevelopedaboutthebehaviorof${zeta(frac{1}{2}+it);tinmathbb{R}}$.128Inparticular,$(RH)$impliesthe(stillunproven)Lindel"ofhypothesis:$
8、zeta(frac{1}{2}+it)
9、=0(oint),$$tarrowinfty$forany$epsilon>0$.Thisconjecturecmbeshowntobeeq
