three notes on connections between the riemann zeta function and probability theory in particular random matrix theory

three notes on connections between the riemann zeta function and probability theory in particular random matrix theory

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时间:2018-02-11

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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

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