On the Riemann zeta-function and the divisor probl

arXiv:math/0404261v2[math.NT]2Jul2004ONTHERIEMANNZETA-FUNCTIONANDTHEDIVISORPROBLEMAleksandarIvi´cAbstract.LetΔ(x)denotetheerrortermintheDirichletdivisorproblem,andE(T)theerrortermintheasymptoticformulaforthemeansquareof|ζ(12+it)|.IfE∗(t)=E(t)−2πΔ∗(t/2π)withΔ∗(x)=−Δ(x)+2Δ(2x)−12Δ(4x),thenweobtainZT0(E∗(t))4dt≪εT16/9+ε.WealsoshowhowourmethodofproofyieldstheboundRXr=1Ztr+Gtr−G|ζ(12+it)|2dt4≪εT2+εG−2+RG4Tε,whereT1/5+ε≤G≪T,Tt1···tR≤2T,tr+1−tr≥5G(r=1,...,R−1).1.IntroductionandstatementofresultsLet,asusual,(1.1)Δ(x)=Xn≤xd(n)−x(logx+2γ−1)−14,and(1.2)E(T)=ZT0|ζ(12+it)|2dt−TlogT2π
+2γ−1,whered(n)isthenumberofdivisorsofn,γ=−Γ′(1)=0.577215...isEuler’sconstant.ThusΔ(x)denotestheerrortermintheclassicalDirichletdivisorproblem,andE(T)istheerrorterminthemeansquareformulafor|ζ(12+it)|.1991MathematicsSubjectClassification.11N37,11M06.Keywordsandphrases.Dirichletdivisorproblem,Riemannzeta-function,meansquareandtwelfthmomentof|ζ(12+it)|,meanfourthpowerofE∗(t).TypesetbyAMS-TEX2A.Ivi´cAninterestinganalogybetweend(n)and|ζ(12+it)|2waspointedoutbyF.V.Atkinson[1]morethansixtyyearsago.Inhisfamouspaper[2],AtkinsoncontinuedhisresearchandestablishedanexplicitformulaforE(T)(seealsotheauthor’smonographs[7,Chapter15]and[8,Chapter2]).Themostsignificanttermsinthisformula,uptothefactor(−1)n,aresimilartothoseinVoronoi’sformula(see[7,Chapter3])forΔ(x).Moreprecisely,in[13]M.JutilashowedthatE(T)shouldbeactuallycomparedto2πΔ∗(T/(2π)),where(1.3)Δ∗(x):=−Δ(x)+2Δ(2x)−12Δ(4x).ThenthearithmeticinterpretationofΔ∗(x)(seeT.Meurman[16])is(1.4)12Xn≤4x(−1)nd(n)=x(logx+2γ−1)+Δ∗(x).Wehavetheexplicit,truncatedformula(seee.g.,[7]or[18])(1.5)Δ(x)=1π√2x14Xn≤Nd(n)n−34cos(4π√nx−14π)+Oε(x12+εN−12)(2≤N≪x).Onealsohas(see[7,eq.(15.68)]),for2≤N≪x,(1.6)Δ∗(x)=1π√2x14Xn≤N(−1)nd(n)n−34cos(4π√nx−14π)+Oε(x12+εN−12),whichiscompletelyanalogousto(1.5).M.Jutila,inhisworks[13]and[14],investigatedboththelocalandglobalbehaviourofE∗(t):=E(t)−2πΔ∗t2π
.Heprovedthemeansquarebound(1.7)ZT+HT−H(E∗(t))2dt≪εHT1/3log3T+T1+ε(1≪H≤T),whichinparticularyields(1.8)ZT0(E∗(t))2dt≪T4/3log3T.Hereandlaterεdenotespositiveconstantswhicharearbitrarilysmall,butarenotnecessarilythesameateachoccurrence.Thebound(1.8)showsthat,ontheOntheRiemannzeta-functionandthedivisorproblem3average,E∗(t)isoftheorder≪t1/6log3/2t,whilebothE(x)andΔ(x)areoftheorder≍x1/4.Thisfollowsfromthemeansquareformulas(seee.g.,[8])(1.9)ZT0Δ2(x)dx=(6π2)−1∞Xn=1d2(n)n−3/2T3/2+O(Tlog4T),and(1.10)ZT0E2(x)dx=23(2π)−1/2∞Xn=1d2(n)n−3/2T3/2+O(Tlog4T).Themeansquareformulas(1.9)and(1.10)alsoimplythattheinequalitiesα1/4andβ1/4cannothold,whereαandβare,respectively,theinfimaofthenumbersaandbforwhichthebounds(1.11)Δ(x)≪xa,E(x)≪xbhold.Forupperboundsonα,βseee.g.,M.N.Huxley[5].Classicalconjecturesarethatα=β=1/4holds,althoughthisisnotoriouslydifficulttoprove.M.Jutila[13]succeededinshowingtheconditionalestimates:iftheconjecturalα=1/4holds,thenthisimpliesthatβ≤3/10.Conversely,β=1/4impliesthatΔ∗(x)≪εxθ+εholdswithθ≤3/10.AlthoughoneexpectsthemaximalordersofΔ(x)andΔ∗(x)tobeapproximatelyofthesameorderofmagnitude,thisdoesseemsdifficulttoestablish.InwhatconcernstheformulasinvolvinghighermomentsofΔ(x)andE(t),wereferthereadertotheauthor’sworks[6],[7]and[10]andD.R.Heath-Brown[4].Inparticular,notethat[10]containsproofsof(1.12)ZT0E3(t)dt=16π4ZT2π0(Δ∗(t))3dt+O(T5/3log3/2T),ZT0E4(t)dt=32π5ZT2π0(Δ∗(t))4dt+O(T23/12log3/2T).InarecentworkbyP.Sargosandtheauthor[12],theasymptoticformulasofK.-M.Tsang[19]forthecubeandthefourthmomentofΔ(x)weresharpenedto(1.13)ZX1Δ3(x)dx=BX7/4+Oε(Xβ+ε)(B0)4A.Ivi´cand(1.14)ZX1Δ4(x)dx=CX2+Oε(Xγ+ε)(C0)withβ=75,γ=2312.Thisimprovesonthevaluesβ=4728,γ=4523,obtainedin[19].Moreover,(1.13)and(1.14)remainvalidifΔ(x)isreplacedbyΔ∗(x),sincetheirproofsusednothingmorebesides(1.5)andtheboundd(n)≪εnε.Hencefrom(1.12)andtheanaloguesof(1.13)–(1.14)forΔ∗(x),weinferthenthat(1.15)ZT0E3(t)dt=B1T7/4+O(T5/3log3/2T)(B10),ZT0E4(t)dt=C1T2+Oε(T23/12+ε)(C10).ThemainaimofthispaperistoprovideanestimatefortheupperboundofthefourthmomentofE∗(t),whichisthefirstnon-trivialupperboundforahighermomentofE∗(t).TheresultisthefollowingTHEOREM1.Wehave(1.16)ZT0(E∗(t))4dt≪εT16/9+ε.Notethatthebounds(1.8)and(1.16)donotseemtoimplyeachother.Fortheproofof(1.16)weshallneedseverallemmas,whichwillbegiveninSection2.TheproofofTheorem1willbegiveninSection3.Finally,inSection4,itwillbeshownhowthemethodofproofofTheorem1cangiveaproofofTHEOREM2.LetT1/5+ε≤G≪T,Tt1···tR≤2T,tr+1−tr≥5G(r=1,···,R−1).Then(1.17)RXr=1Ztr+Gtr−G|ζ(12+it)|2dt!4≪εT2+εG−2+RG4Tε.Theboundin(1.17)easilygivesthewell-knownbound(seeSection4)(1.18)ZT0|ζ(12+it)|12dt≪εT2+ε,duetoD.R.Heath-Brown[3](whohadlog17TinsteadoftheTε-factor).Itisstillessentiallythesharpestresultconcerninghighmomentsof|ζ(12+it)|.Generalsumsofzeta-integralsovershortintervals,analogoustotheoneappearingin(1.17),weretreatedbytheauthorin[9].OntheRiemannzeta-functionandthedivisorproblem52.ThenecessarylemmasLEMMA1(O.Robert–P.Sargos[17]).Letk≥2beafixedintegerandδ0begiven.Thenthenumberofintegersn1,n2,n3,n4suchthatNn1,n2,n3,n4≤2Nand|n1/k1+n1/k2−n1/k3−n1/k4|δN1/kis,foranygivenε0,(2.