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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.<strong>Reference</strong>s [1]N.L.BassilyandI.Katai,Distributionofthevaluesofq-additivefunctionsonpolynomialsequences, ActaMath.Hung.68(1995),353{361. [2]R.BellmanandH.N.Shapiro,Onaprobleminadditivenumbertheory,Ann.Math.49(1948),333{340. [3]L.E.Bush,Anasymptoticformulafortheaveragesumofthedigitsofintegers,Am.Math.Monthly47 (1940),154{156. [4]J.Coquet,Powersumsofdigitalsums,J.NumberTh.22(1986),161{176. [5]H.DelangeSurlafonctionsommatoiredelafonction"SommedeChires",L'Enseignementmath.21 (1975),31{77. [6]M.DrmotaandM.Ska lba,TheparityoftheZeckendorfsum-of-digits-function,preprint. [7]J.M.DumontandA.Thomas,Digitalsummomentsandsubstitutions,ActaArith.64(1993),205{225. [8]J.M.DumontandA.Thomas,Gaussianasymptoticpropertiesofthesum-of-digitsfunctions,J.Number Th.62(1997),19{38. [9]C.-G.Esseen,Fourieranalysisofdistributionfunctions.AmathematicalstudyoftheLaplace-Gaussian law,ActaMath.77(1945),1{125. [10]J.Gajdosik,KombinatorischeFaktorisierungenundZiernentwicklungen,thesis,TUWien,1996. [11]P.J.Grabner,P.Kirschenhofer,H.Prodinger,andR.F.Tichy,Onthemomentsofthesum-of-digits function,in:ApplicationsofFibonacciNumbers5(1993),263{271 [12]P.J.GrabnerandR.F.Tichy,Contributionstodigitexpansionswithrespecttolinearrecurrences,J. NumberTh.36(1990),160{169. [13]P.GrabnerandR.F.Tichy,-Expansions,linearrecurrences,andthesum-of-digitsfunction, manuscriptamath.70(1991),311{324. [14]R.E.KennedyandC.N.Cooper,AnextensionofatheorembyCheoandYienconcerningdigitalsums, FibonacciQ.29(1991),145{149. [15]P.Kirschenhofer,Onthevarianceofthesumofdigitsfunction,LectureNotesMath.1452(1990), 112{116. [16]W.Parry,Onthe-expansionofrealnumbers,ActaMath.Acad.Sci.Hung.,12(1961),401{416. [17]A.PethoandR.F.Tichy,Ondigitexpansionswithrespecttolinearrecurrences,J.NumberTh.33 (1989),243{256. [18]J.Schmid,Thejointdistributionofthebinarydigitsofintegermultiples,ActaArith.43(1984),391{415. [19]W.M.Schmidt,Thejointdistributionthedigitsofcertainintegers-tuples,Studiesinpuremathematics, Mem.ofP.Turan(1983),605{622. [20]H.Trollope,Anexplicitexpressionforbinarydigitalsums,Meth.Mag.41(1968),21{25.
Page 1 and 2: tion(relatedtospecicniteandinniteli
Page 3 and 4: facts(see[5,17,13,15,7,10]).Morepre
Page 5 and 6: impliesj?i(n)=kandconsequentlyl(n)=
Page 7 and 8: THEDISTRIBUTIONOFTHESUM-OF-DIGITSFU
Page 9: forjtj(logN)?2thatZT Hence,(2.2)fol

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