Riemanns-Zeta-Function-and-Beyond

arXiv:math/0309478v1[math.NT]30Sep2003Riemann’sZetaFunctionandBeyondStephenS.Gelbart∗andStephenD.Miller†AbstractInrecentyearsL-functionsandtheiranalyticpropertieshaveassumedacentralroleinnumbertheoryandautomorphicforms.Inthisexpositoryarticle,wedescribethetwomajormethodsforprovingtheanalyticcon-tinuationandfunctionalequationsofL-functions:themethodofintegralrepresentations,andthemethodofFourierexpansionsofEisensteinseries.Specialattentionispaidtotechnicalproperties,suchasboundednessinverticalstrips;theseareessentialinapplyingtheconversetheorem,apow-erfultoolthatusesanalyticpropertiesofL-functionstoestablishcasesofLanglandsfunctorialityconjectures.WeconcludebydescribingstrikingrecentresultswhichrestupontheanalyticpropertiesofL-functions.DedicatedtoIlyaPiatetski-Shapiro,withadmirationContents1Introduction22Riemann’sIntegralRepresentation(1859)52.1MellinTransformsofThetaFunctions...............62.2Hecke’sTreatmentofNumberFields(1916)............82.3Hamburger’sConverseTheorem(1921)...............92.4ThePhragmen-Lindel¨ofPrincipleandConvexityBounds.....123ModularFormsandtheConverseTheorem143.1Hecke(1936).............................143.2Weil’sConverseTheorem(1967)..................183.3MaassForms(1949).........................203.4HeckeOperators...........................204L-functionsfromEisensteinSeries(1962-)214.1Selberg’sAnalyticContinuation...................225GeneralizationstoAdeleGroups24∗PartiallysupportedbytheMinervaFoundation.†SupportedbyNSFgrantDMS-0122799.16Tate’sThesis(1950)267AutomorphicformsonGL(n)287.1Jacquet-Langlands(1970)......................317.2Godement-Jacquet(1972)......................357.3Jacquet-Piatetski-Shapiro-Shalika(1979)..............368Langlands-Shahidi(1967-)378.1AnOutlineoftheMethod......................378.1.1CuspidalEisensteinSeries..................388.1.2LanglandsL-functions....................398.1.3TheConstantTermFormula................398.1.4TheNon-ConstantTerm:LocalCoefficients........398.1.5AnalyticPropertiesandtheQuasi-SplitCase.......408.2GL(2)Example............................418.3BoundednessinVerticalStripsandNon-vanishing........429TheLanglandsProgram(1970-)439.1TheConverseTheoremofCogdell-Piatetski-Shapiro(1999)...439.2ExamplesofLanglandsL-functions:SymmetricPowers.....459.3RecentExamplesofLanglandsFunctoriality(2000-).......469.4ApplicationstoNumberTheory(2001-)..............489.4.1ProgresstowardstheRamanujanandSelbergconjectures499.4.2ThedistributionoftheHeckeeigenvalues,Sato-Tate...511IntroductionIn1859Riemannpublishedhisonlypaper1innumbertheory,ashortten-pagenotewhichdramaticallyintroducedtheuseofcomplexanalysisintothesubject.Riemann’smaingoalwastooutlinetheeventualproofofthePrimeNumberTheoremπ(x)=#{primesp≤x}∼xlogx,x→∞,i.e.limx→∞π(x)logxx=1,bycountingtheprimesusingcomplexintegration(theproofwascompletedhalfacenturylaterbyHadamardanddelaValleePoussin).Alongthispathhefirstshowsthathisζ-function,initiallydefinedinthehalf-planeRe(s)1byζ(s)=∞Xn=11ns=Yp(prime)11−1ps,1See[145],and[36],fortranslations.2hasameromorphiccontinuationtoC.Secondly,heproposeswhathasremainedasperhapsthemost-famousunsolvedproblemofourday:TheRiemannHypothesis:ζ(s)6=0forRes1/2.FormoreonthehistoryofζandRiemann’swork,thereadermayconsult[15,30,36,179].Ourrolehereisnotsomuchtofocusonthezeroesofζ(s),butinsomesenseratheronitspoles.Inparticular,ouremphasiswillbeonexplaininghowweknowthatζ(s)extendsmeromorphicallytotheentirecomplexplane,andsatisfiesthefunctionalequationξ(s):=π−s2Γs2ζ(s)=ξ(1−s).Itisonepurposeofthispapertogivetwoseparatetreatmentsofthisassertion.Wewantalsotocharacterizetheζ-functionassatisfyingthefollowingthreeclassicalproperties(whicharesimplertostateintermsofξ(s),thecompletedζ-function).•Entirety(E):ξ(s)hasameromorphiccontinuationtotheentirecomplexplane,withsimplepolesats=0and1.•FunctionalEquation(FE):ξ(s)=ξ(1−s).•BoundednessinVerticalstrips(BV):ξ(s)+1s+11−sisboundedinanystripoftheform−∞aRe(s)b∞(i.e.ξ(s)isboundedinverticalstripsawayfromitstwopoles).AsecondpurposeistooverviewhowthesetreatmentsandpropertiesextendtoL-functionsassignedtomoregeneralgroupssuchasGL(n),thegroupofinvertiblen×nmatrices(thefunctionζ(s)isattached,weshallsee,toGL(1)).Amajormotivatingfactorforstudyingtheseanalyticconditions(especiallythetechnicalBV)isthattheyhavebecomecrucialinapplicationstotheLanglandsFunctorialityConjectures,wheretheyarepreciselyneededinthe“ConverseTheorem,”whichrelatesL-functionstoautomorphicforms(seeTheorems3.1,3.2,andSection7.3).Moretothepoint,thestudyandusefulnessofL-functionshaspervadedmanybranchesofnumbertheory,whereincomplexanalysishasbecomeanunexpectedly-powerfultool.InSection9wediscusstheconnectionswithsomeofthemostdramaticrecentdevelopments,includingthemodularityofellipticcurves,progresstowardstheRamanujanconjectures,andtheresultsofKimandShahidi.Thetwotreatmentswedescribeare,infact,themajormethodsusedforderivingtheanalyticpropertiesofL-functions.TheTwoMethodsAfirstmethod(Section2)ofanalyticcontinuationisRiemann’s,initiatedin1859.Infact,itwasoneofseveraldifferent,thoughsimilar,proofsk