The true order of the riemann zeta--function

arXiv:math/9809090v1[math.NT]17Sep1998THETRUEORDEROFTHERIEMANNZETA–FUNCTIONONTHECRITICALLINEN.V.KuznetsovUDC511.3+517.43+519.45FortheRiemannzeta-functiononthecriticallinetheterminalestimatehavebeenproved,whichhadbeenconjecturedbyLindel¨ofatthebeginningofthisCenture.TheproofisbasedontheauthorsrelationswhichconnectthebilinearformsoftheeigenvaluesoftheHeckeoperatorswithsumsoftheKloostermansums.Bytheway,itisprovedthatfortheHeckeseries(whichareassociatedwiththeeigenfunctionsoftheautomorphicLaplacian)thenaturalanalogueoftheLindel¨ofconjectureistruealso.Bibl.14.Keywords:theRiemannzeta-function,theKuznetsovtraceformulas,theHeckeseries§0.PRELIMINARIES0.1.Themainresult.OneofthetwomainproblemsinthetheoryoftheRiemannzeta-functionisthequestion:whatistheorderofthisfunctiononthecriticalline?InthisworkIprovethefollowingassertion:theLindel¨ofconjecturefortheRiemannzeta-functionistrue.MoreoverthenaturalanalogueofthisconjecturefortheHeckeseriesistruealso.Itmeansthatwehavethefollowingtwotheorems.Theorem1.Letζ(s)betheRiemannzeta–functionwhichisdefinedforRes1bytwoequalitiesζ(s)=∞Xn=11ns=Yp1−1ps−1(0.1)TypesetbyAMS-TEX34N.V.KUZNETSOVwhereprunsoverallprimes.Thenforanyε0|ζ(1/2+it)|≪tε(0.2)ast→+∞.Theorem2.LetHj(s)betheHeckeserieswhichcorrespondstoj–theigenfunc-tionoftheautomorphicLaplacianforthecaseofthefullmodulargroup.Thenforanyfixedj1andforanyε0wehave|Hj(1/2+it)|≪tε(0.3)ast→+∞.§1.INITIALIDENTITIESToprove(0.2)and(0.3)Iusethefollowingknownfacts.1.1.Thefore–traces.IrestrictmyselfbythecaseofthefullmodulargroupΓ.Letλ0=0λ1...6λj6...aretheeigenvaluesoftheautomorphicLaplacianL=−y2∂2∂x2+∂2∂y2.Itmeansforλ=λjthereisnon–zerosolutionoftheequationLu=λu(1.1)withtheconditionsu(γz)≡uaz+bcz+d=u(z)foranyγ∈Γand(u,u)≡ZΓ\H|u(z)|2dμ(z)∞(heredμ(z)=y−2dxdyistheΓ–invariantmeasureontheupperhalfplaneH;Γ\HisthefundamentaldomainofthefullmodulargroupΓ).Thecontinuousspectrumofthisboundaryproblemliesonthehalf–axisλ1/4;thisspectrumissimpleandthecorrespondingeigenfunctionistheanalyticalcontinuationoftheEisensteinseriesE(z,s).ThisseriesisdefinedbytheequalityE(z,s)=Xγ∈Γ/Γ∞(Imγz)s(1.2)forz∈HandRes1;hereΓ∞isthesyclicsubgroupwhichisgeneratingbythetransformationz7→z+1.Forallswehavetheabsolutelyconvergentseries(theFourierexpansion).THETRUEORDEROFTHERIEMANNZETA–FUNCTION5Theorem1.1.(A.Selberg,S.Chowla[1]).Letz=x+iy,y0;thenE(z,s)=ys+ξ(1−s)ξ(s)y1−s+2ξ(s)Xn6=0τs(n)e(nx)√yKs−1/2(2π|n|y)(1.3)wheree(x)=e2πix,τs(n)=Xd|n,d0|n|d2s−1/2,(1.4)ξ(s)=π−sΓ(s)ζ(2s)(1.5)andKν(·)isthemodifiedBesselfunction(theMcdonaldfunction,whichisde-creasingexponentiallyat+∞)oftheorderν.EacheigenfunctionujofthediscretspectrumhasthesimilarFourierexpansion,butwithoutzerothterm:uj(z)=√yXn6=0ρj(n)e(nx)Kiκj(2π|n|y)(1.6)Hereρj(n)aretheFouriercoefficientsofujandforj1κj=qλj−1/4.(1.7)Notethatinthecaseofthefullmodulargroupλ1≈91.14(itistheresultofthecomputercalculations;see[14],p.650-654).WechooseujberealandeacheigenfunctionisevenoroddunderthereflectionoperatorT−1f
(z)=f(−z);sowehaveT−1uj=εjuj(1.8)withεj=+1orεj=−1.Furthermore,itispossibletaketheseeigenfunctionsbysuchwaythattheyaretheeigenfunctionsforalltheHeckeoperators.Letusdefinethen–thHeckeoperatorT(n)bytheequalityT(n)f
(z)=1√nXad=nd0Xb(modd)faz+bd;(1.9)6N.V.KUZNETSOVthenwehaveforallintegersn,m1T(n)T(m)=Xd|(n,m)Tnmd2=T(m)T(n).(1.10)WetakethecommonsystemoftheeigenfunctionsforLandforallT(n),n1.Inthiscasewehaveforalln1andj1ρj(n)=ρj(1)tj(n),ρj(−n)=εjρj(1)tj(n),(1.11)wheretj(n)aresuch(real)numbersthatT(n)uj=tj(n)uj.(1.12)Wehavethefollowingsplendidexpressionsforthebilinearformsoftheseeigen-valuestj(n).Theorem1.2.(N.Kuznetsov[2],R.Bruggeman[3]).Leth(r)beanevenfunc-tioninrwhichisregularinthestrip|Imr|6ΔforsomeΔ1/2andwichisO|r|−2−δ
forsomeδ0whenr→∞insideofthisstrip.Thenforanyintegersn,m1Xj1αjtj(n)tj(m)h(κj)+1π∞Z−∞τ1/2+ir(n)τ1/2+ir(m)h(r)|ζ(1+2ir)|2dr==12πδn,m∞Z−∞h(u)dχ(u)+Xc11cS(n,m;c)ϕ4π√nmc,(1.13)whereαj=(coshπκj)−1|ρj(1)|2,(1.14)dχ(u)=2πutanh(πu)du,(1.15)SdenotestheKloostermansum,S(n,m;c)=Xa(modc)ad≡1(modc)ena+mdc,(1.16)THETRUEORDEROFTHERIEMANNZETA–FUNCTION7andwithnotation(Jν(·)–theBesselfunctionoftheorderν)k0(x,ν)=12cos(πν)(J2ν−1(x)−J1−2ν(x))(1.17)theweightfunctioninthesumoftheKloostermansumsisdefinedbytheintegraltransformϕ(x)=∞Z−∞k0(x,1/2+ir)h(r)dχ(r).(1.18)Theidentity(1.13)iscalled”theKuznetsovtraceformula”;Ithinkthemorepreferablesay”fore–trace”.IntherealitythefamousSelbergtraceformula(forthefullmodulargroupandforcongruencesubgroups)followsfrom(1.13).Sothesetoftheseidentitieswithalln,m1maybeconsideredasthesetofprimaryequalitiestoconstructthetraceformulae(andalotofothersidentities).1.2.Theregularcase.ThefirstexampleofthesimilaridentitiesistheclassicalPetersonformula.Theorem1.3.(H.Peterson).Letfj,kbetheorthogonalHeckebasisinthespaceMkofcuspformsofanevenweightk,νk=dimMkandtj,k(n)aretheeigen-valuesoftheHeckeoperatorsTk(n):Mk→MkunderthenormalizationTk(n)f
(z)=n(k−1)/2Xd0ad=n1dkXb(modd)faz+bd.(1.19)ThenwehaveνkXj=1kfj,kk−2tj,k(n)tj,k(m)=δn,m+2πi−kXc11cS(n,m;c)Jk−14π√nmc.(1.20)Notethatfork=2,4,6,8,10and14thesumontheleftside(1.20)iszero