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1、ANALYTICMETHODSFORDIOPHANTINEPROBLEMSbyCraigValereSpencerAdissertationsubmittedinpartialfulfillmentoftherequirementsforthedegreeofDoctorofPhilosophy(Mathematics)inTheUniversityofMichigan2008DoctoralCommittee:ProfessorTrevorD.Wooley,UniversityofBristol,Co-ChairEmeritusProfess
2、orDonaldJ.Lewis,Co-ChairProfessorJeffreyC.LagariasAssistantProfessorVilmaM.MesaAssistantProfessorDjordjeMilicevicCraigValereSpencer2008cAllRightsReserveddedicatedtoVernaKisling,MarieSpencer,andthememoryofLyleKislingandEldonSpenceriiACKNOWLEDGEMENTSIwouldliketothankTrevorWool
3、eyforhisadvice,help,funding,andpatience.Withouthisguidance,thisthesiswouldnothavebeenpossible,andIwouldhaveprobablyleftMichiganaftermysecondyear.IconsidermyselfveryfortunatetohavebeenabletolearnhowtousethecirclemethodandpursuemathresearchfromProfessorWooley.Iwouldalsoliketo
4、thankDonaldLewisforsteppinginduringmylastyearofgraduateschoolasmyadvisor.HeprovidedmewewonderfulguidanceandsupportasIwrotethisthesisandappliedforjobs.ThematerialinChaptersIIIandIVofthisthesiswerewrittenwithYu-RuLiu.IwouldliketothankProfessorLiuforworkingwithme,givingmeadvic
5、eaboutresearchandjobs,andfinancingavisittotheUniversityofWaterloo.IwouldalsoliketothankallofmymathinstructorsfromtheUniversityofWisconsin-Stout,theUSA/CanadaMathcamp,CarletonCollege,andtheUniversityofMichigan.Inparticular,IhavebenefitedgreatlybystudyingunderJinhoBaik,BrianCon
6、rad,StephenDeBacker,TinaGarrett,JeromeWilliamHoffman,MarkKrusemeyer,JeffreyLagarias,DjordjeMilicevic,HughMontgomery,DavidRomano,andKannanSoundararajan.Inaddition,discussionsatconferenceswithAndrewGranville,RogerHeath-Brown,HenrykIwaniec,ScottParsell,EmmanuelPeyre,andLynneWall
7、inghavehelpedmeselectdirectionsformyresearch.IamalsoverygratefultoJeffreyLagarias,DonaldLewis,VilmaMesa,DjordjeMilicevic,andiiiTrevorWooleyforservingonmycommittee.IwouldliketothankJamesHirschfeldforprovidingareferenceto[19]andMichaelLipnowskiforfindinganerrorintheoriginalproo
8、fofTheorem4.1.Ihavebeenfortunatetohavetheencouragementofveryunderstandingfriendsan
