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tion(relatedtospecicniteandinnitelinearrecurrences)satisesacentrallimittheorem. Additionallyalocallimittheoremisderived. Abstract.Byusingageneratingfunctionapproachitisshownthatthesum-of-digitsfunc- THEDISTRIBUTIONOFTHESUM-OF-DIGITSFUNCTION MICHAELDRMOTAANDJOHANNESGAJDOSIK
non-negativeintegernhasa(unique)properG-arydigitalexpansion withintegerdigitsj(n)0providedthat LetG=(Gj)j0beastrictlyincreasingsequenceofintegerswithG0=1.Thenevery 1.Introduction
forallk0.Thesum-of-digitsfunctionssG(n)isgivenby sG(n)=Xj0j(n) Xj

approachwasusedin[6].) bytheuseofgeneratingfunctions,whereitisalsopossibletoderivealocallimitlaw.(Asimilar 2Themainpurposeofthispaperistoproveasymptoticnormality(ofthedistributionofXN) MICHAELDRMOTAANDJOHANNESGAJDOSIK
2.1.FiniteRecurrences.Intherstcasewewillmakethefollowingassumptions: niteorinnitelinearrecurrences. 1.Thereexistnon-negativeintegersai,1ir,suchthat(forjr) InthepresentpaperwewilldealwithbasissequencesG=(Gj)j0whichsatisfyspecic Gj=rXi=1aiGj?i: 2.Results
Insection3wewillshowthattheaboveassumptionsimplythatthecharactericticpolynomial 2.gcdfi1:ai6=0g=1. 3.Forallj>rand1k

facts(see[5,17,13,15,7,10]).Moreprecisely,muchmoreisknownaboutthefollowingO(1)terms.Thereforewewillnotpresentaproof.anceofXN(dened)in(1.1).Actuallythisstatementismoreorlessacollectionofwellknown2.3.AsymptoticProperties.Firstwestateatheoremconcerningexpectedvalueandvari- andsoweareinsimilarsituationasabove. THEDISTRIBUTIONOFTHESUM-OF-DIGITSFUNCTION 3
abovetypes.Set Theorem2.1.SupposethatG=(Gj)j0satisesaniteorinnitelinearrecurrenceofthe forzinasucientlysmall(complex)neighbourhoodofz0=1suchthat(1)=.Then andlet1=(z)denotethe(analytic)solutionu=1=(z)oftheequation EXN=1NXn

wehaverPj=1aj=G(1)>1andconsequentlyu01ofmaximalmodulus.Set 4u!1G(u)=1thereuniquelyexistsu0>0withG(u0)=1.SinceGnisstrictlyincreasing MICHAELDRMOTAANDJOHANNESGAJDOSIK
Furthermore,ifjuj=u0butu6=u0thenthegcd-conditiongcdfi1:ai6=0g=1impliesthat Thus,=1=u0>1isasimplerootofP(u). Consequently,therearenorootsofP(u)otherthanwithmodulus. Ifjuj

impliesj?i(n)=kandconsequentlyl(n)=l(m)for0l

wherebj;k=aGjk=jfnj1>>jL?1andej>0.Then Xk0aNkzk=L?1 N=L?1 Xl=0bjl(z)zPl?1 Xl=0elGjl Proof.ByLemma3.2(whichisalsovalidinthecaseofinniterecurrences)wehave aNk=L?1 h=0ehel?1 Xi=0zi: =L?1 Xl=0el?1
Xi=0bjl;k?Pl?1 Xi=0(m

THEDISTRIBUTIONOFTHESUM-OF-DIGITSFUNCTION 7
Proof.Firstly,letGnsatisfyanitelinearrecurrenceoftheabovetype.Then1?G(1;u)=
urP(u?1),whereP(u)isthecharacteristicpolynomial.ThusG(1;?1)=0.Furthermore,since
@
@uG(z;u)

inanopenneighbourhoodoft=0inR.NowsupposethatN=L?1 jL?1andel>0istheG-aryexpansionofN.ThenbyLemma3.6andthetrivialestimate 8el?1 Ps=0eits=el(1+O(t))wehave Xk0aNkeikt=L?1 Xl=0bjl(eit)eitPl?1 MICHAELDRMOTAANDJOHANNESGAJDOSIK
h=0ehel?1 Xs=0eits Pl=0elGjlwithj0>j1>>
andthat Nowobservethat=L?1 Xl=0elGjleijlt?jl2t2=2+itPl?1
HenceEeit(XN?N)=N=e?itN=N1NXk0aNkeikt=N e?itN=N=e?it(=)j1=2 itN=it j1=2 0?1+O(j?1 h=0eheO(t+jlt3+(1?)jl)+O(N(1?)): 0(1+O(j?1 0);
=e?t2=2L?1 Xl=0elGjl Neit=(j1=2 0)?Pl?1 h=0eh?(j0?jl)?t2(jl?j0)=(2j0) 0)):
Let">0bea(small)realnumberandletbedenedbyj?1>j0?j"0j.Then j0?j

forjtj(logN)?2thatZT Hence,(2.2)follows. THEDISTRIBUTIONOFTHESUM-OF-DIGITSFUNCTION ?TN(t) tdt=O(logN)?1=2+"(loglogN): 5.LocalLimitLaw 9
preciseinformationonthebehaviourofbj(z). Proposition5.1.SupposethatG=(Gj)j0satisesaniteorinniterecurrenceofthe abovetypes.Thenthereexist>0and>0suchthat uniformlyforjtj,whereC(z)and(z)areasinProposition4.1,and InordertoprovealocallimitlawforXN,i.e.thesecondpartorTheorem2.2,weneedmore bj(eit)=C(eit)(eit)j+O(1?)j bj(eit)=O(1?)j (5.1)
analytic(andthereforebounded)inthisrangeandweobtain uniformlyforjtj. Proof.Obviously,(5.1)followsfrom(4.1)forsome>0. andjuj1=.Hence,bycontinuitythereexist">0and>0suchthatj1?G(u;eit)j uniformlyfor(real)twithjtjand(complex)uwithjuj1+".Thus,B(eit;u)is Fortheproofof(5.2)wejusthavetoobservethatjG(z;u)j0. pointapproximations. bj(eit)=1
Proposition5.2.Wehave WithhelpofProposition5.1itispossibletoderiveasymptoticexpansionsforbj;kviasaddle =O(1?)j; 2iZ juj=1=+"B(eit;u)du uj+1
Since Proof.WeagainuseCauchy'sformula uniformlyforallj;k0. bj;k=Gj Zjtjjbj(eit)jdt=O(1?)j=O(Gj=j)
p2j2exp?(k?j)2 bj;k=1 2Z?bj(eit)e?iktdt: 2j2+O(j?1=2)
where0

Finally,I1=1 10 =1 2Zjtjj?"Cjeit(j?k)?j2t2=2?1+O(jjt3j+jtj)dt+O(1?)j 2Z1 ?1Cjeit(j?k)?j2t2=2dt+O MICHAELDRMOTAANDJOHANNESGAJDOSIK
=Cj +O p2j2exp?(k?j)2 Zjtjj?"Cje?j2t2=2(jjt3j+jtj)dt!+O(1?)j 2j2+O(j=j) Zjtj>j?"Cje?j2t2=2dt!
=Gj p2j2exp?(k?j)2
Proof.AsintheproofofProposition4.2wesupposethatN=L?1 ThiscompletestheproofofProposition5.2. FinallyProposition5.2andLemma3.6canbeusedtocompletetheproofofTheorem2.2. 2j2+O(Gj=j):
andletbedenedbyj?1>j0?j"0j.Thenby(3.5) jL?1andel>0)istheG-aryexpansionofN.Furthermore,let">0bea(small)realnumber aNk=L?1 Xl=0el?1 Xi=0bjl;k?Pl?1 h=0eh?i Pl=0elGjl(withj0>j1>>
Ifl

THEDISTRIBUTIONOFTHESUM-OF-DIGITSFUNCTION 11
weobtain
aNk=N
p22Nexp?(k?N)2
22Nk?1
Xl=0elGjl
N1+Oj"?1=2
0logj0+O(Gj0=j0)
=N
p22Nexp?(k?N)2
22N+Oj"?1=2
0logj0:
Ifjk?Njj1=2
0logj0thenwehaveforl<
bjl;k?Pl?1
h=0eh?i=Ojlj?1=2
0exp?(logj0)2
42
=O?jlj?1
0
whichnallygives aNk=O(?j0j?1
0+O L?1
Xl=Gjl
j1=2
l!
=O(Gj0=j0):
ThiscompletestheproofofTheorem2.2.References
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