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1、NotAlwaysBuriedDeepSelectionsfromAnalyticandCombinatorialNumberTheoryc2003,2004PaulPollack2003SummerCourseNotesRossSummerMathematicsProgramFirstDraft:June9,2003SecondDraft:November16,2003Lastupdated:May9,2004InmemoryofProfessorArnoldEphraimRoss(1906-20
2、02),whoseexamplecontinuestobeateacherforallofus.PrefaceThegoldin‘themtherehills’isnotalwaysburieddeep.Muchofitiswithineasyreach.Someofitisrightonthesurfacetobepickedupbyanysearcherwithakeeneyefordetailandaneagernesstoexplore.Asinanytreasurehunt,theinvo
3、lvementgrowsasthehuntproceedsandeachsuccesswhethersmallorgreataddsthefuelofexcitementtotheexploration.–ArnoldRoss,Prolog(1978)Thismanuscriptgrewoutofnotesforaneight-weeksummercourseofferedtocounselorsatthe2003RossSummerMathematicsProgram.Asthetitlesugge
4、sts,allofthetopicsdiscussedlieclosetothesurfaceinthattheirstudyrequiresnosignificanttechnicalpreparation.Solidundergraduatecoursesinnumbertheory,abstractalgebraandsingle-variablecalculusshouldenableonetoappreciatebulkofthetext.OutlineofContentsOurfirstch
5、aptertreatselementaryprimenumbertheoryinZ,Z[i]andFq[T].Webeginwithatopicalsurveyofseveralproofsoftheinfinitudeoftheprimes(overZ).WethendigressbrieflytodiscusshowGausswasledtoconjecturetheprimenumbertheorem(whoseproofwedefertoChapter4)beforewetacklesomeof
6、thesimplestestimatesforπ(x),suchasEuler’sestimateπ(x)=o(x).ThenexttwosectionsaredevotedtotheworkofChebyshevandMertens,in-cludingChebyshev’sdeterminationofx/logxasthecorrectorderofmagnitudeofπ(x)andRamanujan’sproofoftheasymptoticBertrand’spostulate.Some
7、analogousresultsinZ[i]arediscussedbeforeweturnourattentiontoprimenumbertheoryintheringofpolynomialsoverafinitefield.Therewefocusonaclassicalanalogoftheprimenumbertheorem,dueessentiallytoGauss,andarecentlyestablishedanalogofthetwinprimeconjecture,due(forq
8、6=3)toC.Hall[Hal03,Corollary19]:ifα∈F∗,whereq>2,thenthereareinfinitelyqmanytwinprimepairsh,h+α∈Fq[T].Ourheuristicapproachtotheprimenumbertheoremalsomotivatesseveralas-yetunprovedconjectures,suchastheGoldbachandtwinprimeconjectures.Suchco
