On the divisor function and the Riemann zeta-funct

arXiv:0707.1756v5[math.NT]8Nov2007ONTHEDIVISORFUNCTIONANDTHERIEMANNZETA-FUNCTIONINSHORTINTERVALSAleksandarIvi´cAbstract.Weobtain,forTε≤U=U(T)≤T1/2−ε,asymptoticformulasforZ2TT(E(t+U)−E(t))2dt,Z2TT(Δ(t+U)−Δ(t))2dt,whereΔ(x)istheerrortermintheclassicaldivisorproblem,andE(T)istheerrorterminthemeansquareformulafor|ζ(12+it)|.UpperboundsoftheformOε(T1+εU2)fortheaboveintegralswithbiquadratesinsteadofsquareareshowntoholdforT3/8≤U=U(T)≪T1/2.TheconnectionbetweenthemomentsofE(t+U)−E(t)and|ζ(12+it)|isalsogiven.Generalizationstosomeothernumber-theoreticerrortermsarediscussed.1.IntroductionPowermomentsrepresentoneofthemostimportantpartsofthetheoryoftheRiemannzeta-functionζ(s),definedasζ(s)=∞Xn=1n−s(σ=ℜes1),andotherwisebyanalyticcontinuation.Ofparticularsignificancearethemomentsonthe“criticalline”σ=12,andavastliteratureexistsonthissubject(seee.g.,themonographs[5],[6],and[23]).Inthispaperweshallbeconcernedwithmomentsoftheerrorfunction(1.1)E(T):=ZT0|ζ(12+it)|2dt−TlogT2π+2γ−1,1991MathematicsSubjectClassification.11M06,11N37.Keywordsandphrases.TheRiemannzeta-function,thedivisorfunctions,powermomentsinshortintervals,upperbounds.TypesetbyAMS-TEX12AleksandarIvi´cwhereγ=−Γ′(1)isEuler’sconstant.Morespecifically,weshallconsiderthemoments(1.2)Z2TT(E(t+G)−E(t−G))kdt(k∈Nfixed),whereG=G(T)is“short”inthesensethatG=O(T)asT→∞andG≫1.Todealwithboundsfortheexpressionsliketheonein(1.2),itseemsconvenienttousealsoresultsonthemomentsofthefunctionE∗(t):=E(t)−2πΔ∗t2π
,whereΔ∗(x):=−Δ(x)+2Δ(2x)−12Δ(4x)=12Xn≤4x(−1)nd(n)−x(logx+2γ−1).Hereasusuald(n)=Pδ|n1isthenumberofpositivedivisorsofn,and(1.3)Δ(x)=Xn≤xd(n)−x(logx+2γ−1)istheerrortermintheclassicalDirichletdivisorproblem.ThefunctionE∗(t)givesaninsightintotheanalogybetweentheDirichletdivisorproblemandthemeansquareof|ζ(12+it)|.Itwasinvestigatedbyseveralauthors,includingM.Jutila[15],whointroducedthefunctionE∗(t),andtheauthor[6]–[8].Amongotherthings,theauthor(op.cit.)provedthatZT0(E∗(t))2dt=T4/3P3(logT)+Oε(T5/4+ε),whereP3isapolynomialofdegreethreeinlogTwithpositiveleadingcoefficient,(1.4)ZT0|E∗(t)|5dt≪εT2+ε,ZT0|E∗(t)|3dt≪εT3/2+ε,andnoneofthesethreeresultsimpliesanyoneoftheothertwo.Fromtheboundsin(1.4)andtheCauchy-Schwarzinequalityforintegralsitfollowsthat(1.5)ZT0|E∗(t)|4dt≪εT7/4+ε.Hereandlaterε(0)denotesarbitrarilysmallconstants,notnecessarilythesameonesateachoccurrence,anda=Oε(b)(sameasa≪εb)meansthattheimpliedOnthedivisorfunctionandtheRiemannzeta-functioninshortintervals3constantdependsonlyonε.Inadditionto(1.2)itmakessensetoinvestigatethemoments(1.6)Z2TT(Δ(t+G)−Δ(t−G))kdt(k∈Nfixed),aswell.TheinterestinthistopiccomesfromtheworkofM.Jutila[12],whoinvestigatedthecasek=2in(1.2)and(1.6).Heprovedthat(1.7)T+HZT(Δ(x+U)−Δ(x))2dx=14π2Xn≤T2Ud2(n)n3/2T+HZTx1/2exp2πiUrnx−12dx+Oε(T1+ε+HU1/2Tε),for1≤U≪T1/2≪H≤T,andananalogousresultholdsalsofortheintegralofE(x+U)−E(x)(theconstantsinfrontofthesumandintheexponentialwillbe1/√2πand√2π,respectively).From(1.7)onededuces(a≍bmeansa≪b≪a)(1.8)ZT+HT(Δ(x+U)−Δ(x))2dx≍HUlog3√TU!forHU≫T1+εandTε≪U≤12√T.In[14]Jutilaprovedthattheintegralin(1.8)is≪εTε(HU+T2/3U4/3)(1≪H,U≪X).Thisboundand(1.8)holdalsofortheintegralofE(x+U)−E(x).FurthermoreJutilaconjecturedthat(1.9)Z2TT(E(t+U)−E(t−U))4dt≪εT1+εU2holdsfor1≪U≪T1/2,andtheanalogousformulashouldholdforΔ(t)aswell.Infact,usingtheideasofK.-M.Tsang[24]whoinvestigatedthefourthmomentofΔ(x),itcanbeshownthatoneexpectstheintegralin(1.9)tobeoforderTU2log6(√T/U).Jutilaalsoindicatedthatthetruthofhisconjecture(1.9)implies(1.10)ZT0|ζ(12+it)|6dt≪εT1+ε.4AleksandarIvi´cThisis(aweakenedformof)thesixthmomentfor|ζ(12+it)|,andthebestknownexponentatpresentontheright-handsideof(1.10)is5/4(see[5],[6]).Inviewofthebound(op.cit.)(1.11)|ζ(12+it)|k≪logtZt+1t−1|ζ(12+ix)|kdx+1,(k∈Nfixed)weactuallyhave,using(1.9)withU=Tεand(1.11)withk=2,(1.12)Z2TT|ζ(12+it)|8dt≪εZ2TTlogT(E(t+Tε)−E(t−Tε))4+Tε dt≪εT1+ε,andtheeighthmomentbound(1.12)isnotablystrongerthan(1.10).ItmayberemarkedthatthefourthmomentsofΔ(x)andE(T)havebeeninvestigatedbyseveralauthors,includingIvi´c–Sargos[11],K.-M.Tsang[24],andW.Zhai[25],[26].2.StatementofresultsOurfirstaimistoderivefrom(1.7)(whenH=T)atrueasymptoticformula.TheresultisTHEOREM1.For1≪U=U(T)≤12√Twehave(c3=8π−2)(2.1)Z2TT(Δ(x+U)−Δ(x))2dx=TU3Xj=0cjlogj√TU+Oε(T1/2+εU2)+Oε(T1+εU1/2),asimilarresultbeingtrueifΔ(x+U)−Δ(x)isreplacedbyE(x+U)−E(x),withdifferentconstantscj.Remark1.ForTε≤U=U(T)≤T1/2−ε(2.1)isatrueasymptoticformula.Corollary1.For1≪U≤12√Twehave(c3=8π−2)(2.2)XT≤n≤2T(Δ(n+U)−Δ(n))2=TU3Xj=0cjlogj√TU+Oε(T1/2+εU2)+Oε(T1+εU1/2),Theformula(2.2)isaconsiderableimprovementoveraresultofCoppola–Salerno[3],whohad(Tε≤U≤12√T,L=logT)(2.3)XT≤n≤2T(Δ(n+U)−Δ(n))2=8π2TUlog3√TU+O(TUL5/2√L).OnthedivisorfunctionandtheRiemannzeta-functioninshortintervals5Corollary2.ForT≤x≤2TandTε≤U=U(T)≤T1/2−εwehave(2.4)Δ(x+U)−Δ(x)=Ωn√Ulog3/2√xUo,E(x+h)−E(x)=Ωn√Ulog3/2√xUo.Theseomegaresults(f(x)=Ω(g(x))meansthatlimx→∞f(x)/g(x)6=0)showthatJutila’sconjecturesmadein[12],namelythat(2.5)Δ(x+U)−Δ(x)≪εxε√U,E(x+U)−E(x)≪εxε√Uforxε≤U≤x1/2−εare(iftrue),closetobeingbestpossible.Thedifficultyoftheseconject