High moments of the Riemann zeta-function

HIGHMOMENTSOFTHERIEMANNZETA{FUNCTIONJ.B.ConreyandS.M.GonekIntroductionAnimportantgoalofnumbertheoriststhiscenturyhasbeentoascertainthebehaviorofmomentsoftheRiemannzeta{function.Theseareimportantforseveralreasons.Therstistheirapplicabilitytothestudyofthedistributionofprimenumbers,oftenthroughzerodensityestimates.Second,theycanbeusedtoestimatethemaximalorderofthezeta{functiononthecriticalline.Notonlyisthisacentralquestioninthetheoryofthezeta{function,butq{analoguescouldallowustoestimatethemaximalsizesofsuchfundamentalarithmeticalobjectsastheclassnumbersofnumbereldsand(assumingthetruthoftheBirchandSwinnerton-Dyerconjecture)theTate{Shafarevichgroupsofellipticcurves.Indeed,thereisacloseconnectionbetweenourquestionandthemaximalsizeofFouriercoecientsofhalf-integralweightcuspforms.TwoofthemostimportantearlyresultswereobtainedbyHardyandLittlewood[HL]in1918andIngham[I]in1926.HardyandLittlewoodprovedthat(1)ZT0j(1=2+it)j2dtTlogTasT!1,andInghamshowedthat(2)ZT0j(1=2+it)j4dt122Tlog4T:Noanalogousformulahasyetbeenprovedforanyhighermoment,anditseemsunlikelythatanywillbeinthenearfuture.Infact,theproblemissointractablethat,untilafewyearsago,noonewasevenabletoproduceaplausibleguessfortheasymptoticmainterm.Recently,however,ConreyandGhosh[CG2]foundaspecialargumentinthecaseofthesixthpowermomentthatledthemtoconjecturethat(3)ZT0j(1=2+it)j6dt429!Yp11p41+4p+1p2!Tlog9T:ResearchofbothauthorswassupportedinpartbytheAmericanInstituteofMathematicsandbygrantsfromtheNSF.TypesetbyAMS-TEX12J.B.CONREYANDS.M.GONEKThemainobjectofthispaperistodescribeanewgeneralapproachwhich,inprinciple,couldproducethecorrectformulaforeveryevenintegermomentIk(T)=ZT0j(1=2+it)j2kdt;andtocarryitoutforboththesixthandeighthpowers.Thisleadstothesamesixthpowermomentconjectureasaboveandlendsadditionalstrongsupporttoitinasensetobedescribedbelow.Fortheeighthpowermoment,weobtainthefollowingnewconjecture.Conjecture1.AsT!1,ZT0j(1=2+it)j8dt2402416!Yp11p91+9p+9p2+1p3!Tlog16T:Wewillalsodescribehowourmethodprovidesinsightintothesizeofhighermomentsofthezeta{function,itsmaximalorderinthecriticalstrip,andthebehaviorofadditivedivisorsums.Whilewritingthispaper,theauthorslearnedthatJ.KeatingandN.Snaith[KS]havemadeahighmomentsconjecturebasedonacompletelydierentapproach.Insteadoftheattackthroughapproximatefunctionalequations,meanvaluetheorems,andadditivedivisorsumsemployedhere,theyproveageneralresultonmomentsofrandommatriceswhoseeigenvalueshaveaGUE(GaussianUnitaryEnsemble)distribution.Ifthezeta{functionismodeledbythedeterminantofsuchamatrix,andtherearereasonstobelieveitis,thenthemomentstheycalculateapplytothezeta{functionaswell.Itisremarkablethatourconjectureandtheirs,whichwestatelater,agreeforthesixthandeighthmoments,anditsuggeststhatbotharelikelytoberight.Webeginbyoutliningthemainideasbehindourapproachstartingwithabriefdiscus-sionofapproximatefunctionalequations.Fors=+itand1,k(s)hastheDirichletseriesexpansionk(s)=1Xn=1dk(n)ns;wheredk(n)isthekthdivisorfunction,whichismultiplicativeanddenedatprimepowersbydk(pj)=k+j1j.Theseriesdoesnotconvergewhen1,butwecanneverthelessapproximatek(s)inthisregionbyasumoftwoDirichletpolynomials.Thisiscalledanapproximatefunctionalequation,anditsprototypeis(4)(s)k=Dk;N(s)+(s)kDk;M(1s)+Ek(s);whereDk;N(s)=NXn=1dk(n)ns;HIGHMOMENTSOFTHERIEMANNZETA{FUNCTION3Ek(s)isanerrorterm,MN=t2k,and(s)=()s1=2(1s2)(s2)isthefactorfromthefunctionalequationforthezeta{function,namely(s)=(s)(1s):Notethatfromthelastequationitfollowsthat(s)satises(s)(1s)=1:Takings=1=2+itin(4),integratingthesquareofthemodulusofbothsides,andassumingthatEk(1=2+it)issucientlysmall,weobtainZ2TTj(1=2+it)j2kdtZ2TTjDk;N(1=2+it)j2dt+Z2TTjDk;M(1=2+it)j2dt+2Z2TT(1=2it)kDk;N(1=2+it)Dk;M(1=2+it)dt:(5)Now(1=2it)=expitlogt2e(1+O(1=t))ast!1,sowendthat(1=2it)k(mn)it=expitlog(t=2e)kmn(1+O(1=t)):Thishasastationaryphaseatt=2(mn)1=k,whichisgenerallyoutsidetheintervalofintegration.Thissuggeststhatthethirdintegralontheright{handsideof(5)issmallerthanthelargerofthersttwo.(ThatitisnolargercanbeseenfromtheCauchy{Schwarzinequality.)ItisprobablyalsothecasethattintheconditionMN=t2kcanbereplacedbyTwhentislarge.Thus,weexpectthatforeverypositiveintegerk,(6)Z2TTj(1=2+it)j2kdtZ2TTjDk;N(1=2+it)j2dt+Z2TTjDk;M(1=2+it)j2dt;where(7)MN=T2kwithM;N1=2;orNT2kifM=0:Infact,usingclassicalmethods,wecanprovethat(6)holdssubjectto(7)whenk=1,andalsowhenk=2providedthatmax(M;N)T.Whenk3,however,theknown4J.B.CONREYANDS.M.GONEKboundsforEk(s)in(4)aretoolargetogive(6),anditisalsodiculttoshowthatthethirdtermin(5)reallyissmallerthantheothertwo.Nevertheless,itispossibletoovercometheseproblemsbyappealingtoamorecomplicatedformoftheapproximatefunctionalequationrstdevelopedbyA.Good[Go]for(s)andfortheL{functionsattachedtocuspforms(whichareanalogousto(s)2).AcarefulapplicationofGood’smethodallowsonetoestablishaformulalike(6)butwiththecoecientsinDk;N(s)andDk;M(s)smoothedbycertainweightfunctions.Wehaveavoidedthisrigorousapproachinordertokeeptheexpositionasstraightforwardaspossi