STOLARSKY'S CONJECTURE AND THE SUM OF DIGITS OF POLYNOMIAL VALUES

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STOLARSKY'S CONJECTURE AND THE SUM OF DIGITS OF POLYNOMIAL VALUES KEVIN G. HARE, SHANTA LAISHRAM, AND THOMAS STOLL Abstract. Let sq(n) denote the sum of the digits in the q-ary expansion of an integer n. In 1978, Stolarsky showed that lim inf n?∞ s2(n2) s2(n) = 0. He conjectured that, as for n2, this limit infimum should be 0 for higher powers of n. We prove and generalize this conjecture showing that for any polynomial p(x) = ahxh + ah?1xh?1 + · · ·+ a0 ? Z[x] with h ≥ 2 and ah > 0 and any base q, lim inf n?∞ sq(p(n)) sq(n) = 0. For any ? > 0 we give a bound on the minimal n such that the ratio sq(p(n))/sq(n) < ?. Further, we give lower bounds for the number of n < N such that sq(p(n))/sq(n) < ?. 1. Introduction Let q ≥ 2 and denote by sq(n) the sum of digits in the q-ary repre- sentation of an integer n. In recent years, much effort has been made to get a better understanding of the distribution properties of sq re- garding certain subsequences of the positive integers.

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Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 17
• Langue: English

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STOLAR D S I K GI Y T ’S S C O O F N P J O E L C Y T N U O R M E IA A L ND VA T L H U E ES SUMOF KEVING.HARE,SHANTALAISHRAM,ANDTHOMASSTOLL Abstract.Lets q (n)denotethesumofthedigitsintheq-ary expansionofanintegern.In1978,Stolarskyshowedthat s 2 (n 2 ) li n m → i ∞ nf=0. )n(s 2 Heconjecturedthat,asforn 2 ,thislimitinfimumshouldbe0 forhigherpowersofn.Weproveandgeneralizethisconjecture showingthatforanypolynomialp(x)=a h x h +a h−1 x h−1 +∙∙∙+ a 0 ∈Z[x]withh≥2anda h >0andanybaseq, liminf s q (p(n)) =0. n→∞ s q (n) Foranyε>0wegiveaboundontheminimalnsuchthatthe ratios q (p(n))/s q (n)<ε.Further,wegivelowerboundsforthe numberofn<Nsuchthats q (p(n))/s q (n)<ε. 1.Introduction Letq≥2anddenotebys q (n)thesumofdigitsintheq-aryrepre- sentationofanintegern.Inrecentyears,muchefforthasbeenmade togetabetterunderstandingofthedistributionpropertiesofs q re- gardingcertainsubsequencesofthepositiveintegers.Wementionthe ground-breakingworkbyC.MauduitandJ.Rivatonthedistribution ofs q ofprimes[9]andofsquares[10].Inthecaseofgeneralpolynomi- alsp(n)ofdegreeh≥2verylittleisknown.Forthecurrentstateof knowledge,werefertotheworkofC.DartygeandG.Tenenbaum[3], whoprovidedsomedensityestimatesfortheevaluationofs q (p(n))in arithmeticprogressions.Theauthors[7]recentlyexaminedthespecial casewhens q (p(n))≈s q (n). K.G.HarewaspartiallysupportedbyNSERC. ComputationalsupportprovidedbyCFI/OITgrant. Th.StollwaspartiallysupportedbyanAPARTgrantoftheAustrianAcademyof Sciences. 1