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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume Number Pages S XX COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN PETER BORWEIN STEPHEN K

K CHOI AND MICHAEL COONS Abstract De64257ne the Liouville function for a subset of the primes by 1 where 8486 is the number of prime factors of coming from counting multiplicity For the traditional Liouville function is the set of all primes De

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume Number Pages S XX COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN PETER BORWEIN STEPHEN K






Presentation on theme: "TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume Number Pages S XX COMPLETELY MULTIPLICATIVE FUNCTIONS TAKING VALUES IN PETER BORWEIN STEPHEN K"— Presentation transcript:

2PETERBORWEIN,STEPHENK.K.CHOI,ANDMICHAELCOONSSimilartotheMobiusfunction,manyinvestigationssurroundingthe{functionconcernthesummatoryfunctionofinitialvaluesof;thatis,thesumL(x):=Xnx(n):Historically,thisfunctionhasbeenstudiedbymanymathematicians,includingLiouville,Landau,Polya,andTuran.RecentattentiontothesummatoryfunctionoftheMobiusfunctionhasbeengivenbyNg[16,17].LargerclassesofcompletelymultiplicativefunctionshavebeenstudiedbyGranvilleandSoundararajan[7,8,9].OneofthemostimportantquestionsisthatoftheasymptoticorderofL(x);moreformally,thequestionistodeterminethesmallestvalueof#forwhichlimx!1L(x) x#=0:Itisknownthatthevalueof#=1isequivalenttotheprimenumbertheo-rem[14,15]andthat#=1 2+"foranyarbitrarilysmallpositiveconstant"isequivalenttotheRiemannhypothesis[3](Thevalueof1 2+"isbestpossible,aslimsupx!1L(x)=p x�:061867,seeBorwein,Ferguson,andMossingho [4]).In-deed,anyresultassertinga xed#2�1 2;1wouldgiveanexpansionofthezero-freeregionoftheRiemannzetafunction,(s),to(s)#.Unfortunately,aclosedformfordeterminingL(x)isunknown.Thisbringsustothemotivatingquestionbehindthisinvestigation:aretherefunctionssimilarto,sothatthecorrespondingsummatoryfunctiondoesyieldaclosedform?ThroughoutthisinvestigationPwilldenotethesetofallprimes.Asananaloguetothetraditionaland ,de netheLiouvillefunctionforAPbyA(n)=(�1) A(n)where A(n)isthenumberofprimefactorsofncomingfromAcountingmulti-plicity.Alternatively,onecande neAasthecompletelymultiplicativefunctionwithA(p)=�1foreachprimep2AandA(p)=1forallp=2A.Everycom-pletelymultiplicativefunctiontakingonly1valuesisbuiltthisway.TheclassoffunctionsfromNtof�1;1gisdenotedF(f�1;1g)(asin[8]).Also,de neLA:=XnxA(n)andRA:=limn!1LA(n) n:Inthispaper,we rstconsiderquestionsregardingthepropertiesofthefunctionAbystudyingthefunctionRA.TherestofthispaperconsidersanextendedinvestigationofthosefunctionsinF(f�1;1g)whicharecharacter{likeinnature(meaningthattheyagreewitharealDirichletcharacteratnonzerovalues).Whilethesefunctionsarenotreallydirectanaloguesof,muchcanbesaidaboutthem.Indeed,wecangiveexactformulaeandsharpboundsfortheirpartialsums.Withinthecourseofdiscussion,theratio(n)=(n)isconsidered.2.PropertiesofLA(x)De nethegeneralizedLiouvillesequenceasLA:=fA(1);A(2);:::g:Theorem1.ThesequenceLAisnoteventuallyperiodic. 4PETERBORWEIN,STEPHENK.K.CHOI,ANDMICHAELCOONSTheorem2.ForthecompletelymultiplicativefunctionA(n),thelimitRAexistsand(4)RA=(Qp2Ap�1 p+1ifPp2Ap�11,0otherwise.Example1.Foranyprimep,Rfpg=p�1 p+1:Tobealittlemoredescriptive,letusmakesomenotationalcomments.DenotebyP(P)thepowersetofthesetofprimes.Notethatp�1 p+1=1�2 p+1:RecallfromabovethatR:P(P)!R,isde nedbyRA:=Yp2A1�2 p+1:ItisimmediatethatRisboundedaboveby1andbelowby0,sothatweneedonlyconsiderthatR:P(P)![0;1].ItisalsoimmediatethatR?=1andRP=0.Remark1.Foranexampleofasubsetofprimeswithmeanvaluein(0;1),considerthesetKofprimesde nedbyK:=pn2P:pn=minq�n3fq2Pgforn2N:Sincethereisalwaysaprimeintheinterval(x;x+x5=8](seeIngham[12]),theseprimesarewellde ned;thatis,pn+1�pnforalln2N.The rstfewvaluesgiveK=f11;29;67;127;223;347;521;733;1009;1361;:::g:Notethatpn�1 pn+1�n3�1 n3+1;sothatRK=Yp2Kp�1 p+1�1Yn=2n3�1 n3+1=2 3:AlsoRK(11�1)=(11+1)=5=6;sothat2 3RK5 6;andRK2(0;1).TherearesomeveryinterestingandimportantexamplesofsetsofprimesAforwhichRA=0.Indeed,resultsofvonMangoldt[18]andLandau[14,15]givethefollowingequivalence.Theorem3.TheprimenumbertheoremisequivalenttoRP=0.Wemaybeabitmorespeci cregardingthevaluesofRA,forA2P(P).Foreach 2(0;1),thereisasetofprimesAsuchthatRA=Yp2Ap�1 p+1= :ThisresultisaspecialcaseofsomegeneraltheoremsofGranvilleandSoundarara-jan[8]. 6PETERBORWEIN,STEPHENK.K.CHOI,ANDMICHAELCOONSTheabovequestioncanbeposedinamoreinterestingfashion.Indeed,notethatforany nitesetofprimesA,wehavethatRA=Yp2Ap�1 p+1=Yp2A(p) (p)=(z) (z)wherez=Qp2Ap,isEuler'stotientfunctionandisthesumofdivisorsfunction.Alternatively,wemayviewthe nitesetofprimesAasdeterminedbythesquare{freeintegerz.Infact,thefunctionffromthesetofsquare{freeintegerstothesetof nitesubsetsofprimes,de nedbyf(z)=f(p1p2pr)=fp1;p2;:::;prg;(z=p1p2pr)isbijective,givingaone{to{onecorrespondencebetweenthesetwosets.Inthisterminology,weaskthequestionas:Question2.Istheimageof(z)=(z):fsquare{freeintegersg!Q\(0;1)asurjection?Thatis,foreveryrationalq2(0;1),isthereasquare{freeintegerzsuchthat(z) (z)=q?Asastart,Theorem4givesanicecorollary.Corollary2.IfSisthesetofsquare{freeintegers,then8:x2R:x=limk!1(nk)S(nk) (nk)9=;=[0;1];thatis,thesetf(s)=(s):s2Sgisdensein[0;1].Proof.Let 2[0;1]andAbeasubsetofprimesforwhichRA= .IfAis nitewearedone,sosupposeAisin nite.WriteA=fa1;a2;a3;:::gwhereaiai+1fori=1;2;3;:::andde nenk=Qki=1ai.Thesequence(nk)satis estheneededlimit.4.Thefunctionsp(n)WenowturnourattentiontoaclassofthosefunctionsinF(f�1;1g)withmeanvalue0.Inparticular,wewishtoexaminefunctionsforwhichasortofRiemannhypothesisholds:functionsforwhichLA(s)=Pn2NA(n) nshasalargezero{freeregion;thatis,functionsforwhichPnxA(n)growsslowly.Indeed,thefunctionsweconsiderinthefollowingsectionshavepartialsumswhichgrowextremelyslow,infact,logarithmically.Becauseofthisslowgrowth,theygiverisetoDirichletserieswhichconvergeintheentirerighthalf{plane(s)&#x]TJ/;ø 9;&#x.962; Tf;&#x 7.1;• 0;&#x Td ;&#x[000;0.Thusourinvestigationdivergessigni cantlyfromfunctionswhichareverysimilarto,butfocusesonthosethatyieldsomeveryinterestingresults.Tothisend,letpbeaprimenumber.RecallthattheLegendresymbolmodulopisde nedasq p=8�&#x]TJ ;� -1;.93; Td;&#x [00;:1ifqisaquadraticresiduemodulop,�1ifqisaquadraticnon-residuemodulop,0ifq0(modp). 8PETERBORWEIN,STEPHENK.K.CHOI,ANDMICHAELCOONSProof.Itisclearthatthetheoremistrueforn=1.Sinceallfunctionsinvolvedarecompletelymultiplicative,itsucestoshowtheequivalenceforallprimes.Notethat(q)=�1foranyprimeq.Nowifn=p,thenk=1and(�1)1p(p)0p(p)=(�1)(1)(1)=�1=(p):Ifn=q6=p,then(�1)0p(q)0p(q)=q p�q p=�q2 p=�1=(q);andsothetheoremisproved.TomirrortherelationshipbetweenLand,denotebyLp(n),thesummatoryfunctionofp(n);thatis,de neLp(n):=nXk=1p(k):ItisquiteimmediatethatLp(n)isnotpositive2forallnandp.To ndanexampleweneedonlylookatthe rstfewprimes.Forp=5andn=3,wehaveL5(3)=5(1)+5(2)+5(3)=1�1�1=�10:Indeed,thenextfewtheoremsaresucienttoshowthatthereisapositivepro-portion(atleast1=2)oftheprimesforwhichLp(n)0forsomen2N.Theorem7.Letn=a0+a1p+a2p2+:::+akpkbethebasepexpansionofn,whereaj2f0;1;2;:::;p�1g.Thenwehave(6)Lp(n):=nXl=1p(l)=a0Xl=1p(l)+a1Xl=1p(l)+:::+akXl=1p(l):Herethesumoverlisregardedasemptyifaj=0.InsteadofgivingaproofofTheorem7inthisspeci cform,wewillproveamoregeneralresultforwhichTheorem7isadirectcorollary.Tothisend,letbeanon-principalDirichletcharactermodulopandforanyprimeqlet(7)f(q):=(1ifp=q,(q)ifp6=q.Weextendftobeacompletelymultiplicativefunctionandget(8)f(plm)=(m)forl0andp-m.Theorem8.LetN(n;l)bethenumberofdigitslinthebasepexpansionofn.ThennXj=1f(j)=p�1Xl=0N(n;l)0@Xml(m)1A: 2ForthetraditionalL(n),itwasconjecturedbyPolyathatL(n)0foralln,thoughthiswasproventobeanon-trivialstatementandultimatelyfalse(SeeHaselgrove[11]). 10PETERBORWEIN,STEPHENK.K.CHOI,ANDMICHAELCOONSProof.We rstobservefrom(5)thatif0rp,thenrXl=1p(l)=rXl=1l p:Fromtheorem7,nXl=1p(l)=a0Xl=1p(l)+a1Xl=1p(l)+:::+akXl=1p(l)=a0Xl=1l p+a1Xl=1l p+:::+akXl=1l pbecauseallajarebetween0andp�1.Theresultthenfollows.Corollary4.Forn2N,wehave0L3(n)[log3n]+1:Proof.ThisfollowsfromTheorem9,Application1,andthefactthatthenumberof1'sinthebasethreeexpansionofnis[log3n]+1.Asafurtherexample,letp=5.Corollary5.ThevalueofL5(n)isequaltothenumberof1'sinthebase5ex-pansionofnminusthenumberof3'sinthebase5expansionofn.Alsoforn1,jL5(n)j[log5n]+1:Recallfromabove,thatL3(n)isalwaysnonnegative,butL5(n)isn't.AlsoL5(n)=kforthe rsttimewhenn=50+51+52+:::+5kandL5(n)=�kforthe rsttimewhenn=350+351+352+:::+35k.Remark2.Thereasonforspeci cpvaluesintheproceedingtwocorollariesisthat,ingeneral,it'snotalwaysthecasethatjLp(n)j[logpn]+1.Wenowreturntoourclassi cationofprimesforwhichLp(n)0foralln1.De nition3.DenotebyL+,thesetofprimespforwhichLp(n)0foralln2N.Wehavefound,bycomputation,thatthe rstfewvaluesinL+areL+=f3;7;11;23;31;47;59;71;79;83;103;131;151;167;191;199;239;251:::g:Byinspection,L+doesn'tseemtocontainanyprimesp,withp1(mod4).Thisisnotacoincidence,asdemonstratedbythefollowingtheorem.Theorem10.Ifp2L+,thenp3(mod4).Proof.Notethatifp1(mod4),thena p=�a pforall1ap�1,sothatp�1 2Xa=1a p=0: 12PETERBORWEIN,STEPHENK.K.CHOI,ANDMICHAELCOONS6.ConcludingremarksThroughoutthispaper,wewereinterestedinestimatesconcerningthepartialsumsofmultiplicativefunctionsinF(f�1;1g).Animportantquestionis:whatcanbesaidaboutthegrowthof Pnxf(n) foranyfunctionf2F(f�1;1g)?ThisquestiongoesbacktoErd}os.Indeed,Erd}os[5]states:Finally,Iwouldliketomentionanoldconjectureofmine:letf(n)=1beanarbitrarynumber{theoreticfunction[notnecessarilymultiplicative].Isittruethattoeverycthereisadandanmsothat mXk=1f(kd) �c?Ihavemadenoprogresswiththisconjecture.Concerningthisconjecture,headdsin[6]thatthebestwecouldhopeforisthatmaxmdn mXk=1f(kd) �clogn:Weremarkthatthesequestionscanalsobeaskedforfunctionsf(n)whichtakekthrootsofunityasvaluesratherthanjust1.However,verylittleisyetknownforthiscase.IfwerestrictErd}os'questiontotheclassofcompletelymultiplicativefunctions,itmaybepossibletoprovideananswer.Whichwouldbeinteresting.Recallthatforpanoddprimeandpacharacter{likefunction,wehavemaxNx XnNp(n) logx;sothattheclassofcharacter{likefunctionssatis esErd}os'conjecture.Thesefunctionsarepartofalargerclassoffunctionscalledautomaticfunctions(see[1]).LetT=(t(n))n1beasequencewithvaluesfroma niteset.De nethek{kernelofTasthesetT(k)=f(t(kln+r))n0:l0and0rklg:Givenk2,wesayasequenceTisk{automaticifandonlyifthek{kernelofTis nite.Asageneralizationofourresultconcerningcharacter{likefunctions,weaskthefollowingquestion.Question3.Letf2F(f�1;1g)bek{automaticforsomek2,andsupposethatPnxf(n)=o(x).ThenisittruethatmaxNx PnNf(n) logx?References1.J.-P.AlloucheandJ.Shallit.Automaticsequences.CambridgeUniversityPress,Cambridge,2003.2.PaulT.BatemanandHaroldG.Diamond,Analyticnumbertheory,WorldScienti cPublish-ingCo.Pte.Ltd.,Hackensack,NJ,2004,Anintroductorycourse.3.P.Borwein,S.Choi,B.Rooney,andA.Weirathmuller,TheRiemannHypothesis:AResourcefortheAcionadoandVirtuosoAlike,CMSBooksinMathematics,vol.27,Springer,NewYork,2008.4.P.Borwein,R.Ferguson,andM.J.Mossingho ,SignchangesinsumsoftheLiouvillefunc-tion,Math.Comp.(2007),toappear.