On the mean values the Riemann zeta-function in sh

arXiv:0803.0132v2[math.NT]25Jul2008ONTHEMEANVALUESOFTHERIEMANNZETA-FUNCTIONINSHORTINTERVALSAleksandarIvi´cAbstract.Itisprovedthat,forTε≤G=G(T)≤12√T,Z2TT“I1(t+G)−I1(t)”2dt=TG3Xj=0ajlogj“√TG”+Oε(T1+ε+T1/2+εG2)withsomeexplicitlycomputableconstantsaj(a30)where,forfixedk∈N,Ik(t):=Ik(t,G)=1√πZ∞−∞|ζ(12+it+iu)|2ke−(u/G)2du.ThegeneralizationstothemeansquareofI1(t+U,G)−I1(t,G)over[T,T+H]andtheestimationofthemeansquareofI2(t+U,G)−I2(t,G)arealsodiscussed.1.IntroductionThemeanvaluesoftheRiemannzeta-functionζ(s),definedasζ(s)=∞Xn=1n−s(σ=ℜes1)andotherwisebyanalyticcontinuation,occupyacentralplaceinthetheoryofζ(s).Ofparticularsignificanceisthemeansquareonthe“criticalline”σ=12,andavastliteratureexistsonthissubject(seee.g.,themonographs[3],[4],and[18]).Oneusuallydefinestheerrorterminthemeansquareformulafor|ζ(12+it)|as(1.1)E(T):=ZT0|ζ(12+it)|2dt−TlogT2π+2γ−1,1991MathematicsSubjectClassification.11M06,11N37.Keywordsandphrases.TheRiemannzeta-function,themeanvaluesinshortintervals,upperbounds.TypesetbyAMS-TEX12AleksandarIvi´cwhereγ=−Γ′(1)=0.5772156649...isEuler’sconstant.Moregenerally,onehopesthatforafixedkthefunction(E(T)≡E1(T)inthisnotation)(1.2)Ek(T):=ZT0|ζ(12+it)|2kdt−TPk2(logT)(k∈N)representstheerrortermintheasymptoticformulaforthe2k-thmomentof|ζ(12+it)|,wherePℓ(z)isasuitablepolynomialinzoddegreeℓ.Thisisknown,besidesthecasek=1,onlyinthecasek=2(seee.g.,[4],[17]),andanyfurtherimprovementwouldbeofgreatsignificance,inviewofnumerousapplicationsofpowermomentsof|ζ(12+it)|.BymeansofrandommatrixtheoryplausiblevaluesofthecoefficientsofthepolynomialPk2(z)thatoughttobestandingin(1.2)aregivenbyJ.B.Conreyetal.[2].However,thesevaluesarestillconjectural.AsforexplicitformulasforEk(T)andrelatedfunctions,webeginbymentioningthefamousformulaofF.V.Atkinson[1].Let0AA′beanytwofixedconstantssuchthatATNA′T,letd(n)=Pδ|n1bethenumberofdivisorsofn,andfinallyletN′=N′(T)=T/(2π)+N/2−(N2/4+NT/(2π))1/2.Then(1.3)E(T)=X1(T)+X2(T)+O(log2T),where(1.4)X1(T)=21/2(T/(2π))1/4Xn≤N(−1)nd(n)n−3/4e(T,n)cos(f(T,n)),(1.5)X2(T)=−2Xn≤N′d(n)n−1/2(log(T/(2πn))−1cos(Tlog(T/(2πn))−T+π/4),with(βjareconstants)(1.6)f(T,n)=2Tarsinhpπn/(2T)
+p2πnT+π2n2−π/4=−14π+2√2πnT+16√2π3n3/2T−1/2+β5n5/2T−3/2+β7n7/2T−5/2+...,(1.7)e(T,n)=(1+πn/(2T))−1/4n(2T/πn)1/2arsinh(pπn/(2T))o−1=1+O(n/T)(1≤nT),OnthemeanvaluestheRiemannzeta-functioninshortintervals3andarsinhx=log(x+√1+x2).Atkinson’sformulacameintoprominenceseveraldecadesafteritsappearance,andincitedmuchresearch(seee.g.,[3],[4]forsomeofthem).Thepresenceofthefunctiond(n)in(1.4)andthestructureofthesumΣ1(T)pointouttheanalogybetweenE(T)andΔ(x),theerrortermintheclassicaldivisorproblem,definedasΔ(x)=Xn≤xd(n)−x(logx+2γ−1).Thisanalogywasinvestigatedbyseveralauthors,mostnotablybyM.Jutila[13],[15],andthenlaterbytheauthor[6]–[11].Jutila[13]provedthat(1.8)T+HZTΔ(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).Hereandlaterε(0)denotesarbitrarilysmallconstants,notnecessarilythesameonesateachoccurrence,whilea≪εbmeansthattheimplied≪–constantdependonε.From(1.8)onededuces(a≍bmeansa≪b≪a)(1.9)ZT+HTΔ(x+U)−Δ(x)2dx≍HUlog3√TU!forHU≫T1+εandTε≪U≤12√T.In[14]Jutilaprovedthattheintegralin(1.9)is≪εTε(HU+T2/3U4/3)(1≪H,U≪T).Thisboundand(1.9)holdalsoforthemeansquareintegralofE(x+U)−E(x).FurthermoreJutilaconjecturedthat(1.10)Z2TTE(t+U)−E(t−U)4dt≪εT1+εU2holdsfor1≪U≪T1/2,andtheanalogousformulashouldholdforΔ(t)aswell.Infact,usingtheideasofK.-M.Tsang[19]whoinvestigatedthefourthmomentofΔ(x),itcanbeshownthatoneexpectstheintegralin(1.10)tobeoftheorder4AleksandarIvi´cTU2log6(√T/U).Asshownin[11],thetruthofJutila’sconjecture(1.10)impliesthehithertounknowneighthmomentbound(1.11)ZT0|ζ(12+it)|8dt≪εT1+ε,whichwouldhaveimportantconsequencesinmanyproblemsfrommultiplicativenumbertheory,suchasboundsinvolvingdifferencesbetweenconsecutiveprimes.DespiteseveralresultsonE2(T)(seee.g.,[17]),noexplicitformulaisknownforthisfunction,whichwouldbeanalogoustoAtkinson’sformula(1.3)-(1.7).Thisisprobablyduetothecomplexityofthefunctioninquestion,anditisevennotcertainthatsuchaformulaexists.However,whenoneworksnotdirectlywiththemomentsof|ζ(12+it)|,butwithsmoothedversionsthereof,thesituationchanges.Let,fork∈Nfixed,(1.12)Ik(t,G):=1√πZ∞−∞|ζ(12+it+iu)|2ke−(u/G)2du(1≪G≪t).Y.Motohashi’smonograph[17]containsexplicitformulasforI1(t)andI2(t)insuitablerangesforG=G(t).TheformulaforI2(t)involvesquantitiesfromthespectraltheoryofthenon-EuclideanLaplacian(seeop.cit.).Letasusual{λj=κ2j+14}∪{0}bethediscretespectrumofthenon-EuclideanLaplacianactingonSL(2,Z)–automorphicforms,andαj=|ρj(1)|2(coshπκj)−1,whereρj(1)isthefirstFouriercoefficientoftheMaasswaveformcorrespondingtotheeigenvalueλjtowhichtheHeckeseriesHj(s)isattached.Then,forT1/2log−DT≤G≤T/logT,andforanarbitraryconstantD0,Motohashi’sformulagives(1.13)G−1I2(T,G)=O(log3D+9T)+π√2T∞Xj=1αjH3j(12)κ−1/2jsinκjlogκj4eTexp−14GκjT2.Forourpurposestherangeforwhich(1.13)holdsisnotlargeenough.Weshalluseam