2012 Update May 25 2013 From highly composite numbers to transcendental number theory by Michel Waldschmidt Abstract In a wellknown paper published in 1915 in the Proceedings of the London Mathematical Society Srinivasa Ramanujan de64257ned and st ID: 68885
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numberisapositiveintegernforwhichthereexists"�0suchthat,forallk�1,(n) n1+"(k) k1+"Here,isthefunctionsumofdivisors:(n)=Xdjnd:Thesequenceofcolossallyabundantnumbers(referenceA004490in[32])startswith2;6;12;60;120;360;2520;5040;55440;720720;1441440;4324320;21621600;:::Thesuccessivequotientsare:3;2;5;2;3;7;2;11;13;2;3;5;:::Intheirpaper[1],AlaogluandErd}oswrite:...thismakesqxrational.Itisverylikelythatqxandpxcannotberationalatthesametimeexceptifxisaninteger.Thiswouldshowthatthequotientoftwoconsecutivecolossallyabundantnumbersisaprime.Atpresentwecannotshowthis.ProfessorSiegelhascommunicatedtoustheresultthatqx,rxandsxcannotbesimul-taneouslyrationalexceptifxisaninteger.Hencethequotientoftwoconsecutivecolossallyabundantnumbersiseitheraprimeortheproductoftwodistinctprimes.Thisistheoriginoftheproblemthatwenowconsider.2FourexponentialsConjectureandsixexpo-nentialsTheoremIfpandqaredistinctprimesandifxisarealnumbersuchthatpx=randqx=sareintegers,thenwehavex=logr logp=logs logqandthematrixlogplogqlogrlogshasrank1.Onedoesnotknowanexamplewherethishappenswithoutxbeinganinteger.Moregenerally,ifwereplacetheassumptionthatpandqaredis-tinctprimesbytheassumptionthattheyaretwomultiplicativelyindependentpositivenumbers,thentheexpectedconclusionisthatxshouldberational.Recallthattwopositivenumberspandqaremultiplicativelyindependentiftherelationpaqb=1withaandbrationalintegersimpliesa=b=0.Hencethe2 iftisanirrationalnumber,atleastoneofthethreenumbers2t,2t2,2t3istranscendental.Incasetisalgebraic,thesethreenumbersaretranscendentalbyGel'fond{Schneider'sTheorem.Iftisatranscendentalnumberanda,b,carepositiveintegerswithb6=c,thenofatleastoneofthenumbers2ta;2tb;2tc;2ta+b;2ta+cistranscendental.Forinstanceifaandbarepositiveintegers,atleastoneofthenumbers2ta;2tb;2ta+b;2t2a+b:istranscendental.AconsequenceofSchanuel'sConjecturewouldbethatallnumbers2n(n1)aretranscendental.AspecialcaseofthefourexponentialsConjectureisthatatleastoneofthetwonumbers2,22istranscendental.AccordingtothesixexponentialsTheorem,atleastoneofthethreenumbers2,22,23istranscendental.Algebraicapproximationsto2khavebeeninvestigatedbyT.N.Shorey[31]andS.Srinivasan[33,34].Seealso[13].Upperboundsforthenumberofalgebraicnumbersamong2k,(1kN)havebeenobtainedbyS.Srinivasan[34].Seealso[20,21].3FiveexponentialsTheorem,strongfourexpo-nentialsConjectureandstrongsixexponen-tialsTheoremTheveexponentialsTheoremwasprovedin1986[39]Cor.2.2.FiveexponentialsTheorem(Exponentialform).Letx1;x2betwoQ{linearlyindependentcomplexnumbersandy1;y2bealsotwoQ{linearlyindependentcomplexnumbers.Thenatleastoneofthe5numbersex1y1;ex1y2;ex2y1;ex2y2;ex2=x1istranscendental.Thenextresultisstronger:FiveexponentialsTheorem(Matrixform).LetMbea23matrixwhoseentriesareeitheralgebraicnumbersorlogarithmsofalgebraicnumbers.Assumethatthethreecolumnsarelinearlyindependentover Qandthatthetworowsarealsolinearlyindependentover Q.ThenMhasrank2.Wededucetheexponentialformfromthematrixformbyconsideringthematrixlog 11log 121log 21log 22log =x1y1x1y21x2y1x2y2x2=x1:4 4LowerboundfortherankofmatriceswithentrieslogarithmsofalgebraicnumbersTheconclusionsofthepreviousresultsarethatsomematriceshaverank�1.ThesixexponentialsTheoremhasbeengeneralizedin1980[36]inordertoproducelowerboundsfortherankofmatriceswithentriesinLofanysize.Undersuitableassumptions,therankrofsuchad`matrixsatisesrd` d+`Hence,whend=`,rd 2;whichishalfofwhatisexpected.Onecannotexpecttoreachthemaximalrankifoneonlyassumesthatthecolumnsandtherowsarelinearlyindependent:thematrix0@0log2�log3�log20log5log3�log501Ahasrank2only.Themainresultof[36](seealsoTheorem12.17of[42])statesthatad`matrixofranknwithentriesinLisQ{equivalenttoablocmatrixABC0whereCisad0`0matrixwithd0�0andn d`0 d0+`0AconsequenceofthisresultistheanswertoaquestionfromA.Weil[37]:ifthevaluesofaHeckeGrossencharacterarealgebraic(resp.inanumbereld),thenthecharacterisoftypeA(resp.A0).AnotherconsequenceistheanswerbyD.Royin[26]toaquestionraisedbyJ.-L.Colliot-Thelene,D.CorayandJ.-J.Sansuc:givenanumbereldkwithagroupofunitsofrankr,thesmallestpositiveintegermforwhichthereexistsanitelygeneratedsubgroupofrankmofkhavingadenseimagein(R Qk)underthecanonicalembeddingisr+2.Thereisaversionfornonarchimedeanvaluations,whichimpliesthelowerboundrpr=2forthep{adicrankrpoftheunitsofanalgebraicnumbereld,intermsoftherankrofthegroupofunits[36,38],whileLeopoldt'sConjecturepredictsrp=r.SeealsotheresultswhichareprovedbyM.Laurent[8]andthosewhichareclaimedin[14,15]byP.Mihailescu.Also,theultrametrictranscendenceresulthasapplicationsto`{adicrep-resentations[6].LetKbeanumbereld,andGK=Gal( Q=K)theGalois6 Again,thisnumberrsrt(M)doesnotdependonthechoiceofthebasisfM1;:::;Msg.Further,foramatrixMwithentriesinL,thetwodenitionsofrsrt(M)comingfromtheinclusionLeL,coincide.In[27],D.Royprovesthattherankr(M)ofMsatises:r(M)1 2rsrt(M):6Schanuel'sConjectureSchanuel'sConjecture[7]statethatLetx1;:::;xnbeQ{linearlyindependentcomplexnumbers.Thenatleastnofthe2nnumbersx1;:::;xn;ex1;:::;exnarealgebraicallyindependent.OneofthemostimportantandopenspecialcasesofSchanuel'sconjectureistheconjectureonalgebraicindependenceoflogarithmsofalgebraicnumbers:if1;:::;narelinearlyindependentlogarithmsofalgebraicnumbers,thenthesenumbersarealgebraicallyindependent.Sofar,itisnotevenknownifthereexisttwologarithmsofalgebraicnum-berswhicharealgebraicallyindependent.Baker'sresultprovidesasatisfactoryanswerforthelinearindependenceofsuchnumbersovertheeldofalgebraicnumbers.Buthesaysnothingaboutalgebraicindependence.Eventhenon{existenceofnon{trivialquadraticrelationsamonglogarithmsofalgebraicnum-bersisnotyetestablished.AccordingtothefourexponentialsConjecture,anyquadraticrelation(log 1)(log 4)=(log 2)(log 3)istrivial:eitherlog 1andlog 2arelinearlydependent,orelselog 1andlog 3arelinearlydependent.Howeversomethingisknownontheconjectureofalgebraicindependenceoflogarithmsofalgebraicnumbers.Insteadoftakinglinearlyindependentlog-arithmsofalgebraicnumbers1;:::;nandaskingaboutthenon{vanishingofvaluesP(1;:::;n)ofpolynomialsP2Z[X1;:::;Xn],D.Roylooksatthisquestionfromanotherpointofview:startingwithanonzeropolynomialP2Z[X1;:::;Xn],heinvestigatesthetuples(1;:::;n)2Lnwhosecompo-nentsarelogarithmsofalgebraicnumberssuchthatP(1;:::;n)=0.Moregenerally,heremarkedthattheconjectureofalgebraicindependenceofloga-rithmsofalgebraicnumbersisequivalenttothenextstatement:ifVisananealgebraicsubvarietyofCn,thenthesetLn\ViscontainedintheunionoflinearsubspacesofCnrationaloverVcontainedinV.In[27],heprovesspecialcasesofthisstatement.Seealsothepaper[4]byS.Fischler.TheconjectureonalgebraicindependenceoflogarithmsofalgebraicnumberswouldsolvethequestionoftherankofmatriceshavingentriesinthespaceQ[Lspannedby1andthelogarithmsofalgebraicnumbers.Conversely,ithasbeenprovedbyD.RoythattheconjectureonalgebraicindependenceoflogarithmsisequivalenttotheconjecturethatherankofamatrixwithentriesinQ[Lisequaltoitsstructuralrank.Thekeylemma([24,27]{seealso[42]x12.1.5)isthatifkisaeldandP2k[X1;:::;Xn]apolynomialinnvariables,thenthereexistsasquarematrixM,whoseentriesarelinearcombinationsof1;X1;:::;Xnwithcoecientsink,suchthatPisthedeterminantofM.8 ApromisingstrategyforprovingSchanuel'sConjecturehasbeendevisedbyD.Royin[28](seealsox15.5.3of[42]).HeproposesanewconjecturewhichheshowstobeequivalenttoSchanuel'sConjecture.Inaseriesofrecentpapers,heprovedspecialcasesofhisconjecture{seeforinstance[29].7ElllipticfourexponentialsConjectureThequestionofalgebraicindependenceoflogarithmsofalgebraicnumberscanbegeneralizedbyreplacingthemultiplicativegroupbyotheralgebraicgroup,likeanellipticcurve(earlyexamplesarein[18]whichisexpandedin[43])oracommutativegroupvariety.AnexampleofsuchageneralizationoccursinconnectionwithaproblemofK.MahleretYu.V.ManinonthetranscendenceofJ(q)foralgebraicq=e2i .ThisproblemhasbeensolvedbyK.Barre{Siriex,G.Diaz,F.GramainandG.Philibertin1996[3];seealso[44].MixedfourexponentialsTheorem.Letlog bealogarithmofanon{zeroalgebraicnumber.LetZ!1+Z!2bealatticeassociatedwithaWeierstrassellipticcurvehavingalgebraicinvariantsg2,g3.Thenthematrix0@!1log !22i1Ahasrank2.Hereisastrongerstatement.Let}beaWeierstraellipticfunctionwithalgebraicinvariantsg2,g3andEbethecorrespondingellipticcurve.DenotebyLEthesetofu2Cwhicheitherarepolesof}oraresuchthat}(u)isalgebraic.MixedfourexponentialsConjecture.Letu1andu2betwoelementsinLEandlog 1,log 2betwologarithmsofalgebraicnumbers.AssumefurtherthatthetworowsofthematrixM=u1log 1u2log 2arelinearlyindependentoverQ.ThenthematrixMhasrank2.8DensityquestionsAsshownin[41],thesequestionsarerelatedwithaproblemofB.Mazur[9,10,11,12]onthedensityofrationalpointsonvarieties.Seealso[16,17].Justtogiveanexample,apositiveanswertothenextquestionwouldfollowfromthefourexponentialsConjecture(see[43]).For =a+bp 22Q(p 2),write =a�bp 2.Dene 1:=2p 2�1; 2:=3p 2�1; 3:=4p 2�1;9 [13]M.Mignotte&M.Waldschmidt{Approximationsimultaneedevaleursdelafonctionexponentielle,CompositioMath.,34(1977),127{139.[14]P.Mihailescu{OnLeopoldt'sconjectureandsomeconsequences,arXiv:0905.1274Date:Fri,8May200914:52:57GMT(16kb)http://arxiv.org/abs/0905.1274http://arxiv.org/abs/0905.1274v2http://arxiv.org/abs/0905.1274v3http://arxiv.org/abs/0905.1274v4[15]|,ApplicationsofBakerTheorytotheConjectureofLeopoldt,http://arxiv.org/abs/0909.2738http://arxiv.org/abs/1105.5989SNOQIT:SeminarNotesonOpenQuestionsinIwasawaTheory[16]D.Prasad{AnanalogueofaconjectureofMazur:aquestioninDio-phantineapproximationontori,inContributionstoautomorphicforms,geometry,andnumbertheory,JohnsHopkinsUniv.Press,Baltimore,MD(2004),699{709.[17]G.Prasad&A.S.Rapinchuk{Zariski{densesubgroupsandtranscen-dentalnumbertheory,Math.Res.Lett.12(2005),n2{3,239{249.[18]K.Ramachandra{Contributionstothetheoryoftranscendentalnumbers(I),ActaArith.,14(1968),65{72;(II),id.,73{88.[19]|,Lecturesontranscendentalnumbers,TheRamanujanInstituteLectureNotes,vol.1,TheRamanujanInstitute,Madras(1969).[20]K.Ramachandra&R.Balasubramanian{Transcendentalnumbersandalemmaincombinatorics,Proc.Sem.CombinatoricsandApplications,IndianStat.Inst.,(1982),57{59.[21]K.Ramachandra&S.Srinivasan{AnotetoapaperbyRamachandraontranscendentalnumbers,Hardy{RamanujanJournal,6(1983),37{44.[22]S.Ramanujan{Onhighlycompositeandsimilarnumbers,Proc.LondonMath.Soc.(1915)(2)14,347{409[23]|,Highlycompositenumbers,annotatedbyJean{LouisNicolas&GuyRobin.RamanujanJ.1,n2,119{153(1997).[24]D.Roy{Matricesdontlescoecientssontdesformeslineaires,SeminairedeTheoriedesNombres,Paris1987{88,273{281,Progr.Math.,81,BirkhauserBoston,MA(1990).[25]D.Roy{Matriceswhosecoecientsarelinearformsinlogarithms,J.NumberTheory41(1992),n1,22{47.11 [41]|,Densitedespointsrationnelssurungroupealgebrique,Experiment.Math.3(1994),n4,329{352.[42]|,Diophantineapproximationonlinearalgebraicgroups,Grund.derMath.Wiss.326,Springer{Verlag,Berlin(2000).[43]|,OnRamachandra'scontributionstotranscendentalnumbertheory,Ra-manujanMathematicalSocietyLectureNotesSeries2(2006),175{179.[44]|,EllipticFunctionsandTranscendence,DevelopmentsinMathematics17,SurveysinNumberTheoryx7,SpringerVerlag(2008),143{188.MichelWALDSCHMIDTUniversitePierreetMarieCurie(Paris6)TheoriedesNombresCasecourrier2474PlaceJussieu,F{75252PARISCedex05,FRANCEmiw@math.jussieu.frhttp://www.math.jussieu.fr/miw/13