Three Lectures on the Riemann Zeta-Function

arXiv:math/0401126v1[math.NT]13Jan2004THREELECTURESONTHERIEMANNZETA-FUNCTIONS.M.GONEKDEPARTMENTOFMATHEMATICSUNIVERSITYOFROCHESTERROCHESTER,N.Y.14627U.S.A.IntroductionTheselecturesweredeliveredatthe“InternationalConferenceonSubjectsRelatedtotheClayProblems”heldatChonbukNationalUniversity,Chonju,KoreainJuly,2002.MyaimwastogivemathematiciansandgraduatestudentsunfamiliarwithanalyticnumbertheoryanintroductiontothetheoryoftheRiemannzeta–functionfocusing,inparticular,onthedistributionofitszeros.ProfessorY.YildirinoftheUniversityofAnkara,whoalsodeliveredasetoflecturesattheconference,concentratedonthedistributionofprimenumbers.AfewgeneralremarksaboutthelecturesareinorderbeforeIsummarizetheircontents.First,sinceIcouldonlycoverasmallpartofthesubjectinthetimealloted,mychoicesaboutwhattoincludeandexcludewerenecessarilypersonal.Second,Ihaveglossedoveranumberoftechnicaldetailsinordertokeepthefocusonthemainideas.Finally,thereisalmostnothingnewinthelectures.TheexceptionisthedescriptionofanewrandommatrixmodelduetoC.Hughes,J.Keating,andtheauthorattheendofthethirdlecture.IshouldalsoaddthatthismanuscriptisaverycloserecordofthelecturesIdeliveredandthis,Ithink,accountsforthesomewhatbreezystyle.InthefirstlectureIpresentedthebasicbackgroundmaterialonthezeta–function,sketchedaproofofthePrimeNumberTheorem,explainedhowtheRiemannHypothesis(RH)comesintothepicture,andbrieflysummarizedtheevidenceforit.InthesecondlectureIwantedtoexplainhowonestudiesthedistributionofthezerosandchosemean–valueestimatesasaunifyingtheme.Idescribedwhatmean–valueestimatesare,gaveseveralexamples,andexplainedinageneralwaytheirconnectionwiththezeros.Ithensketchedtheideasbehindtwoapplications–themostprimitivezero–densityestimate(duetoH.BohrandE.Landau)andtheproofofN.Levinson’sfamousresultthatatleastone–thirdofthezerosofthezeta–functionlieonthecriticalline.BothresultswerecitedinLectureIasevidencefortheRiemannHypothesis.IhadalsointendedtopresenttheconditionalTheworkoftheauthorwassupportedinpartbyagrantfromtheNationalScienceFoundation.12S.M.GONEKDEPARTMENTOFMATHEMATICSUNIVERSITYOFROCHESTERROCHESTER,N.Y.14627U.S.A.resultofJ.B.Conrey,A.Ghosh,andtheauthorthatmorethanseventypercentofthezerosaresimple,buttherewasnotenoughtime.However,Ihaveincludedthatapplicationhere.ThethirdlecturebeganwiththeobservationthattheRiemannHypothesisdoesnotanswerallourquestionsabouttheprimes;onealsoneedsdetailedinformationabouttheverticaldistributionofthezerosonthecriticalline.IthenpresentedH.Montgomery’spioneeringworkonthepaircorrelationofthezeros.IntheremainderofthelectureIstatedtheGUEhypothesisanddescribedthemostrecentworkonmodelingthezeta–functionbycharacteristicpolynomialsofrandommatricesfromtheCircularUnitaryEnsemble(CUE).Forthosewishingtostudythezeta–functioninmoredepth,themostimportantbooksarebyH.Davenport[D],H.M.Edwards[E],A.E.Ingham[I2],A.Ivic[Iv],andE.C.Titchmarsh[T1],[T2].ForabackgroundinrandommatrixtheorythereadershouldconsultM.L.Mehta[M]andP.Deift[Df].ItakethisopportunitytothanktheorganizersandthemanyotherfineKoreanmathematiciansIgottomeetforthefirsttimeattheconference.Thanksalsotothemathematiciansandstudentswhosowarmlyhostedusvisitingmathematiciansandmadetheconferencesuchanenjoyableandmemorableone.THREELECTURESONTHERIEMANNZETA-FUNCTION3LectureITheZeta–Function,PrimeNumbers,andtheZerosAlthoughmostmathematiciansareawarethattheprimenumbers,theRiemannzeta–function,andthezerosofthezeta–functionareintimatelyconnected,veryfewknowwhy.InthisfirstlectureIwilloutlinethebasicpropertiesofthezeta–function,sketchaproofoftheprimenumbertheorem,andshowhowthelocationofthezerosofthezeta–functiondirectlyinfluencesthedistributionoftheprimes.IwillthenexplainwhytheRiemannHypothesis(RH)isimportantandtheevidenceforit.1TheRiemannzeta-functionTheRiemannzeta–functionisdefinedbytheDirichletseriesζ(s)=∞Xn=1n−s,whichcanalsobewrittenζ(s)=Yp(1+p−s+p−2s+···)=Yp(1−p−s)−1,wheres=σ+itisacomplexvariable.Weimmediatelyseethatthezeta–functionisbuiltoutoftheprimenumbers.Observethattheseriesandproductbothconvergeabsolutelyinthehalf–planeσ1.TheirequalityinthisregionmayberegardedasananalyticequivalentoftheFundamentalTheoremofArithmetic.FortheFundamentalTheoremassuresusthateachtermn−sintheseriesoccursonce,andonlyonce,amongthetermsresultingfrommultiplyingouttheEulerproduct.Conversely,ifweknowtheequalityofthesumandproduct,theFundamentalTheoremfollows.Fromtheequalityofthesumandproductwecanalsodeducethewellkownfactthatthereareaninfinitenumberofprimes.Foriftherewerenot,theproductwouldremainboundedasσ→1+,whereasweknowthatthesumtendstoinfinity.SincenofactorintheEulerproductequalszerowhenσ1,wededucethatζ(s)6=0whenσ1.Also,sincetheseriesconvergeabsolutelywhenσ1,itconvergesuniformlyincompactsubsetsthere.Itfollowsthatζ(s)isanalyticinthehalf–planeσ1.Themostfundamentalpropertiesofthezeta–functionare:(1)Analyticcontinuation:ζ(s)hasananalyticcontinuationtoCexceptforasimplepoleats=1.(2)Functionalequation:Thezeta–functionsatisfiesthefunctionalequationπ−s/2Γ(s/2)ζ(s)=π−(1−s)/2Γ((1−s)/2)ζ(1−s).(3)Trivialzeros:Theonlyzerosofζ(s)inσ0aresimpleonesats=−2,