INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS

INSTITUTEOFPHYSICSPUBLISHINGJOURNALOFPHYSICSA:MATHEMATICALANDGENERALJ.Phys.A:Math.Gen.36(2003)2945–2962PII:S0305-4470(03)40067-XRandommatrixtheoryandthezerosofζ
(s)FrancescoMezzadriSchoolofMathematics,UniversityofBristol,UniversityWalk,Bristol,BS81TW,UKE-mail:[e-mail]30July2002,infinalform5November2002Published12March2003Onlineatstacks.iop.org/JPhysA/36/2945AbstractWestudythedensityoftherootsofthederivativeofthecharacteristicpolynomialZ(U,z)ofanN×NrandomunitarymatrixwithdistributiongivenbyHaarmeasureontheunitarygroup.BasedonpreviousrandommatrixtheorymodelsoftheRiemannzetafunctionζ(s),thisisexpectedtobeanaccuratedescriptionforthehorizontaldistributionofthezerosofζ
(s)totherightofthecriticalline.WeshowthatasN→∞thefractionoftherootsofZ
(U,z)thatlieintheregion1−x/(N−1)
|z|1tendstoalimitfunction.Wederiveasymptoticexpressionsforthisfunctioninthelimitsx→∞andx→0andcomparethemwithnumericalexperiments.PACSnumbers:02.10.Yn,02.10.De1.IntroductionWestudythedensityoftherootsofZ
(U,z)=ddzdet(Iz−U)=ddzN
j=1(z−eiθj)z∈CwhereUisarandomN×Nunitarymatrix,withrespecttothecircularunitaryensemble(CUE)ofrandommatrixtheory(RMT).OurmainmotivationistoinvestigatethehorizontaldistributionofthezerosofthederivativeoftheRiemannzetafunction.Thezetafunctionisdefinedbyζ(s)=∞n=11nsσ=Re(s)1andhasananalyticcontinuationintherestofthecomplexplaneexceptforasimplepoleats=1.Thereareinfinitelymanynon-trivialsolutionstotheequationζ(s)=0inthestrip0σ1;theRiemannhypothesis(RH)statesthattheyalllieonthecriticallineσ=1/2.Theinterestinthehorizontaldistributionofthezerosofζ
(s)ismotivatedbyitsconnection0305-4470/03/122945+18$30.00©2003IOPPublishingLtdPrintedintheUK29452946FMezzadriwithRH.In1934Speiser[1]showedthatRHisequivalenttoζ
(s)havingnozerosintheregion0σ1/2.Furthermore,uptonowthemostefficientwaysofcomputingthefractionofthezerosoftheRiemannzetafunctiononthecriticallinearebasedonwhatisknownasLevinson’smethod[2];itturnsoutthatthezerosofζ
(s)closetothecriticallinehaveasignificanteffectontheefficiencyofthistechnique[3],thereforeitisimportanttoknowhowtheyaredistributed.LevinsonandMontgomery[4]provedaquantitativerefinementofSpeiser’stheorem,namelythatζ(s)andζ
(s)haveessentiallythesamenumberofzerostotheleftofσ=1/2,andshowedthatasT→∞,whereTistheheightonthecriticalline,apositiveproportionofthezerosofζ
(s)areintheregionσ12+(1+)loglogTlogT0.SubsequentimprovementsofLevinsonandMontgomery’sresults,firstbyConreyandGhosh[3],thenbyGuo[5],Soundararajan[6]andrecentlybyZhang[7]haveestablishedthatatypicalzeroofζ
(s)tendstobemuchclosertothecriticallineandthatconditionallyonRHapositiveproportionlieintheregionσ12+ClogTforsomepositiveconstantC.Theirdistribution,however,isstillunknown.Otherresultsonthezerosofζ
(s)canbefoundin[8].Overthepastthirtyyears,overwhelmingevidencehasbeenaccumulatedwhichsuggeststhatthelocalcorrelationsofthenon-trivialzerosofζ(s)coincide,asT→∞,withthoseoftheeigenvaluesofHermitianmatricesoflargedimensionsfromtheGaussianunitaryensemble(GUE)[9].AsN→∞,theGUEstatisticsareinturnthesameasthoseofthephasesoftheeigenvaluesofN×Nunitarymatrices,onthescaleoftheirmeandistance2π/N,averagedovertheCUEensemble.Morerecently,however,itwasrealizedthatRMTnotonlydescribeswithhighaccuracythedistributionoftheRiemannzeros,butalsoprovidestechniquestomakepredictionsandcomputationsabouttheRiemannzetafunctionandcertainclassesofL-functionsthatpreviousmethodshadnotbeenabletotackle.ThisstartedwiththeworkofKeatingandSnaith[10]onmomentsoftheRiemannzetafunctionandotherL-functions.Theirkeyobservationwasthatthelocallydeterminedstatisticalpropertiesofζ(s)highupthecriticallinecanbemodelledbycharacteristicpolynomialsZ(U,z)ofrandomunitarymatricesU.Inthismodelthetwoasymptoticparameters,Tforζ(s)andNforU,arecomparedbysettingthedensitiesofthezerosofζ(s)andoftheeigenvaluesofUequal,i.e.N=logT2π.Thisapproachhassincebeenextremelysuccessful[11].Followingthesameideas,inthispaperwesuggestthatthedensityρ(z)oftherootsofZ
(U,z)willaccuratelydescribethedistributionofthezerosofζ
(s).Aclassicaltheoremincomplexanalysisstatesthatifp(z)isapolynomial,thentherootsofp
(z)thatarenotrootsofp(z)lieallintheinteriororontheboundaryofthesmallestconvexpolygoncontainingthezerosofp(z)(see,e.g.,[12]).Therefore,sincetheeigenvaluesofaunitarymatrixhavemodulusone,thesolutionsoftheequationZ
(U,z)=0thatarenotzerosofZ(U,z)areallinsidetheunitcircle.Ifs=1/2+it+u,t∈R,denotesthepointatwhichζ
(s)isevaluated,thentheregionofCtotherightofthecriticallineismappedinsidetheunitcirclebytheconformalmappingz=e−u.Thus,theradialdensity2π0|z|ρ(z)dφRandommatrixtheoryandthezerosofζ
(s)2947becomestheanalogueofthehorizontaldistributionofthezerosofζ
(s)totherightofthelineσ=1/2.However,insteadofρ(z),itturnsouttobemoreconvenienttoconsiderIp(x),thefractionoftherootsintheannulus1−x/(N−1)
|z|1,wherexisthescaleddistancefromtheunitcircle.OurmainresultsconcerntheasymptoticsofIp(x).WeshowthatasNincreasestherootsofZ
(U,z)approachtheunitcircle,andIp(x)tendstoafunctionindependentofN.Furthermore,weobtainthefollowingasymptoticsasN→∞:Ip(x)∼1−1/xx→∞