TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume Number Pages S XX COMPLETELY - PDF document

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;1
wouldgiveanexpansionofthezero-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).Foreach2(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(p1p2


pr)=fp1;p2;:::;prg;(z=p1p2


pr)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.Let2[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)1
p(p)
0p(p)=(�1)
(1)
(1)=�1=(p):Ifn=q6=p,then(�1)0
p(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=3
50+3
51+3
52+:::+3
5k.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.