Journal of Number Theory

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1 Journal of Number Theory ) Contents lists available at SciVerse ScienceDirect Journal of Number Theory Digit sums of binomial sums Arnold Knopfmacher a,florianluca b,a, a The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, P.O. Wits 2050, South Africa b Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P , Morelia, Michoacán, Mexico article info abstract Article history: Received 30 March 2011 Accepted 11 July 2011 Availableonline4October2011 Communicated by Michael A. Bennett Keywords: Sum of digits Binomial coefficients Linear forms in logarithms Let b 2 be a fixed positive integer and let Sn) be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of Sn) in base b is at least c 0 log n/log log n), where c 0 is some positive constant depending on b and on the sequence of binomial ) sums. Our results include middle binomial 2n coefficients n and Apéry numbers An. The proof uses a result of McIntosh regarding the asymptotic expansions of such binomial sums as well as Baer s theorem on lower bounds for nonzero linear forms in logarithms of algebraic numbers Elsevier Inc. All rights reserved. 1. Introduction Let r := r 0, r 1,...,r m ) be fixed nonnegative integers and put Sn) := n ) r0 ) n n + r1 ) n + rm m for n = 0, 1,... 1) In what follows, we put r := r 0 + +r m. We assume that r 0 > 0. When r = 1), we simply get that Sn) = n ) n = 2 n for all n 0. 2) * Corresponding author at: Instituto de Matemáticas, Universidad Nacional Autonoma de México, C.P , Morelia, Michoacán, Mexico. addresses: Arnold.Knopfmacher@wits.ac.za A. Knopfmacher), fluca@matmor.unam.mx F. Luca) X/$ see front matter 2011 Elsevier Inc. All rights reserved. doi: /j.jnt

2 A. Knopfmacher, F. Luca / Journal of Number Theory ) When r = 2), we get that Sn) = n ) 2 n = b n for all n 0, 3) where b n = ) 2n n is the middle binomial coefficient, while when r = 2, 2), we get that n ) 2 ) n n + 2 Sn) = = A n for all n 0, 4) where A n is the nth Apéry number. Let b 2 be any integer and put s b m) for the sum of the base b digits of the positive integer m. Lower bounds for s b m) when m runs through the members of a sequence with some interesting combinatorial meaning have been investigated before. For example, it follows from a result of Stewart [8]; see also [1] for a slightly more general result), that the inequality log n s b F n )>c 1 log logn 5) holds for all n 3 with some positive constant c 1 depending on b, where F n is the nth Fibonacci number given by F 0 := 0, F 1 := 1 and F n+2 := F n+1 + F n for all n 0. In [2], it is shown that the inequality s b n!)>c 2 log n 6) holds for all n 1, where c 2 is some positive constant depending on b. In [4], it is shown that the inequality s b b n ) εn) log n 7) holds on a set of n of asymptotic density 1, where εn) is any function tending to zero when n tends to infinity. A similar result holds with b n replaced by c n := b n /n + 1) for n = 0, 1,..., which is the nth Catalan number. In [3], it is shown that there is some positive constant c 3 depending on b such that the inequality s b A n )>c 3 log n log logn ) 1/4 8) holds on a set of n of asymptotic density 1. The proofs of such results use a variety of methods from number theory, such as elementary methods, sieve methods, lower bounds for nonzero linear forms in logarithms of algebraic numbers and the subspace theorem of Evertse, Schlicewei and Schmidt. Here, we add on the literature on the topic and prove the following theorem. Theorem 1. For any r 1), there exists a positive constant c 0 := c 0 b, r) depending on both b and r such that the inequality ) log n s b Sn) > c0 log logn holds on a set of positive integers n of asymptotic density 1.

3 326 A. Knopfmacher, F. Luca / Journal of Number Theory ) When r = 1), thensn) = 2 n,soeitherb is a power of 2, in which case s b 2 n ) = O1), orb is not a power of 2, in which case it follows from Stewart s result [8] that the inequality s b 2 n )> c 4 log n/ log log n holds for all sufficiently large positive integers n with some positive constant c 4 depending on b. It is easy to see that particular cases of Theorem 4 such as 3) and 4) are improvements upon estimates 7) and 8). While strictly speaing the Catalan numbers c n arenotoftheformsn) for any particular choice of r, the conclusion of Theorem 4 applies to them also as we shall indicate at the proof of Theorem 4. Features of the proof are a complete asymptotic expansions for Sn) due to McIntosh [7] and a lower bound for a nonzero linear form in logarithms of algebraic numbers due to Matveev [6]. Until now, and in what follows, we use c 0, c 1,... for computable positive constants that appear increasingly throughout the paper and which might be absolute or depend on the number b and the vector r. We use the Landau symbol O and the Vinogradov symbols, and with their usual meanings. Recall that A = OB), A B and B A are all equivalent to the fact that the inequality A cb holds with some constant c. The constants implied by these symbols in our arguments might depend on the number b and the vector r. Furthermore,A B means that both A B and B A hold. 2. Preliminary results We start with McIntosh s asymptotic formula for Sn) see [7]). Lemma 2. For each nonnegative integer p, Sn) = μ n+1/2 1 + ν2πλn) r 1 p =1 ) ) R 1 n + O, n p+1 9) where 0 <λ<1 is defined by 1 = μ = ν = m j=0 1 + jλ) j λ1 + j 1)λ) j 1 m ) r 1 + jλ j, 1 + j 1)λ j=0 m j=0 ) r j, r j 1 + j 1)λ)1 + jλ), and each R is a rational function of the exponents r 0, r 1,...,r m and λ. We shall also need a result of Matveev [6] from transcendental number theory. But first, some notation. For an algebraic number η having F X) := a 0 d X η i) ) Z[X] i=1

4 A. Knopfmacher, F. Luca / Journal of Number Theory ) as minimal polynomial over the integers, the logarithmic height of η is defined as hη) := 1 d log a 0 + { log max η i), }) 1. d i=1 With this notation, Matveev [6] proved a deep theorem a particular case of which is the following. Lemma 3. Let K be a real field of degree D, η 1 > 0, η 2 > 0 be elements of K, andb 1,b 2 be nonzero integers. Put B := max{ b 1, b 2 } and Let A 1,A 2 be real numbers such that Λ := b 1 log η 1 b 2 log η 2. A j max { Dhη j ), log η j, 0.16 }, for j = 1, 2. Then there exists an absolute constant c 5 such that if Λ 0, then log Λ > c 5 D log D)1 + log B)A 1 A The proof of Theorem 4 We let x be a large positive real number. We let δ>0 be sufficiently small to be determined later, and let { ) N δ x) := n [x, 2x): s b Sn) <δ log x log log x }. 10) We need to show that if δ is sufficiently small, then #N δ x) = ox) as x, for after this the conclusion of Theorem 4 will follow by replacing x by x/2, then by x/4, and so on, and summing up the resulting estimates. For n N δ x), wewrite Sn) = d 1 b n 1 + d 2 b n 2 + +d s b n s, 11) where d 1,...,d s {1,...,b 1} and n 1 > n 2 > > n s.weletk be some large number depending on b and r to be determined later, and we put t := tn) for the smallest index i {1, 2,...,s 1} such that b n i n i+1 > n K if it exists and set t := s otherwise. From the definition of tn), we see immediately that Sn) = ) d 1 b n 1 + +d t b n t O n K )) )) := b mn) 1 Dn) 1 + O, x K 12) where m = mn) := n t and Dn) := d 1 b n 1 n t + d 2 b n 2 n t + +d t. Let D δ x) be the subset of all possible values for Dn). Let us find an upper bound for the cardinality of this set. Observe first that ) log x t s s b Sn) δ log log x.

5 328 A. Knopfmacher, F. Luca / Journal of Number Theory ) Next, observe that for a fixed t, the vector d 1,...,d t ) canbechoseninb 1) t < x δ )/ log log x ways. Finally, we also have that n i 1 n i K log n K log2x) for i = 2,...,t. Thus, the vector of numbers n 1 n 2,...,n t 1 n t ) canbechoseninatmost K log2x) ) t 1 ) log x < exp δ log log log x K log2x) )) < x 2δ ways, where the last inequality holds for all sufficiently large x. Moreover, the vector of neighboring differences n 1 n 2,n 2 n 2,...,n t 1 n t ) determines uniquely the vector of exponents n 1 n t,n 2 n t,...,n t 1 n t ) appearing in the base b representation of D via the fact that Thus, we get easily that t 1 n i n t = n j n j+1 ) for i = 1,...,t 1. j=i ) δ log x #D δ x) x δ /log log x) x 2δ < x 3δ, 13) log log x where the last inequality holds provided that x is sufficiently large. We now compare relations 12), 9) and get that where we put Taing logarithms in Eq. 14), we get that We now write where )) μ n f n) = b mn) 1 Dn) 1 + O, x K 14) ) K μ 1 R f n) := ν2πλn) r n 15) 1 n log μ + log f n) mn) log Dn) = O N δ x) = D D δ x) x K ). N δ,d x), 16) N δ,d x) := { n N δ x): Dn) = D }.

6 A. Knopfmacher, F. Luca / Journal of Number Theory ) Assume that N δ,d x) has T := T δ,d x) elements and let them be n 1 < n 2 < < n T.Letn = n i for some i T 1 and write n i+1 =: n +. Then taing the difference of the relations 16) in n and n +, we get log μ + log f n + ) log f n) ) mn + ) mn) ) 1 = O x K ). 17) If r = 1, then r = 1), a case which is excluded. Thus, r > 1 and, in particular, f n) is not constant. By elementary calculus, we get that [ ] f n + ) f n) = d dz f z) z=ζ [n,n+], 18) x r+1)/2 where ζ is some point in [n,n + ] the existence of which is guaranteed by the Intermediary Value Theorem. Observe also that μ > 1becauseSn) tends to infinity with n. Putm := mn + ) mn) and Λ := log μ m. 19) We tae K := r + 1)/2 +1. If Λ = 0, we then get by estimates 17) and 18), that x 1 r+1)/2 x, K which is impossible for 1forlargex because K >r + 1)/2. Thus, Λ 0. Then estimates 17) and 18) again show that Λ. 20) x r+1)/2 Since r 2 and x, it follows that the right-hand side above is o1) as x.thus, mn + ) mn) c 6 as x, where c 6 := log μ)/). Now Matveev s result Lemma 3 applied to Λ, with the choices of parameters η 1 := μ, η 2 := b, b 1 :=, b 2 := mn + ) mn), together with the fact that B = max{, mn + ) mn) }, shows that there exists a positive constant c 7 depending on b and r actually, it depends on b and the height of the algebraic number μ, butthislastparameter depends on r), such that the inequality Λ > 1 c 7 21) holds for all sufficiently large x. We may assume that c 7 >r 1)/2. Putting together 20) and 21), we get that 1 c 7 x r+1)/2, giving x c 8, where c 8 := r + 1)/21 + c 7 )) 0, 1). Since the distance between any two elements of N δ,d x) is x c 8, and this set is contained in [x, 2x), it follows that #N δ,d x) x c 9,

7 330 A. Knopfmacher, F. Luca / Journal of Number Theory ) Fig. 1. The binary sum of digits for the Apéry numbers. where we put c 9 := 1 c 8. This was for an arbitrary D D δ x). Thus, by estimate 13), #N δ x) = D D δ x) #N δ,d x) x c 9 #D δ x)<x 3δ+c 9, provided that x is large enough. The exponent of x in the right-most bound above is < 1 provided that δ<1 c 9 )/3 = c 8 /3. So, we can tae c 0 = δ := c 8 /4 and complete the proof of this theorem. 4. Comments and conjectures What was important for our argument was not the actual formula for Sn) but the fact that it is an integer which has an asymptotic expansion given as in 9) with some algebraic number μ > 1 and some positive integer r > 1. In particular, it also applies to Catalan numbers c n.wegivenofurther details. The binary expansions of Catalan numbers were also studied in [5]. Aproblemoffurtherinterestwouldbetofindthetruerateofgrowthofs b Sn)) for large n. In Fig.1weshowforthecaseb = 2 and Sn) equal to the Apéry numbers, a plot of s b Sn))/n for n = 1,...,1000 and on the right, a plot of the average digit sum 1 n n i=1 s bsi)) for n = 1,...,1000. Based on these and similar computations in other bases b, we are lead to the following conjectures: Conjecture 4. For any r 1), there exists a positive constant c 0 := c 0 b, r) depending on both b and r such that the inequality holds for all positive integers n. s b Sn) ) > c0 n Conjecture 5. For any r 1), there exists a positive constant c 1 := c 1 b, r) depending on both b and r such that the limit holds as n. 1 n Furthermore, we believe that c 1 = log μ n ) s b Si) c1 i=1 ) b 1 4 ). Conjecture 5 with this value of c 1 would follow assuming that the digits of Sn) in base b are uniformly distributed. From our calculations see Fig. 1), it seems that Conjecture 5 might hold for the Apéry numbers in base 2 with c 1 = , which is

8 A. Knopfmacher, F. Luca / Journal of Number Theory ) in agreement with our prediction since for the Apéry numbers we have μ = 1+ 2) 4,so logμ log 2 ) 1 4 ) = log1+ 2) log 2 = Acnowledgments This paper was written while F.L. was in sabbatical from the Mathematical Institute UNAM from January 1 to June 30, 2011 and supported by a PASPA fellowship from DGAPA. References [1] F. Luca, Distinct digits in base b expansions of linear recurrence sequences, Quaest. Math ) [2] F. Luca, The number of non-zero digits of n!, Canad. Math. Bull ) [3] F. Luca, I.E. Shparlinsi, On the g-ary expansions of Apéry, Motzin and Schröder numbers, Ann. Comb ) [4] F. Luca, I.E. Shparlinsi, On the g-ary expansions of middle binomial coefficients and Catalan numbers, Rocy Mountain J. Math., in press. [5] F. Luca, P.T. Young, On the binary expansion of the odd Catalan numbers, in: F. Luca, P. Stănică Eds.), Proceedings of the XIVth International Conference on Fibonacci Numbers and Their Applications, Morelia, Mexico, July 5 9, 2010, in press. [6] E.M. Matveev, An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers, II, Izv. Ross. Aad. Nau Ser. Mat ) in Russian); translation in: Izv. Math ) [7] R. McIntosh, An asymptotic formula for binomial sums, J. Number Theory ) [8] C.L. Stewart, On the representation of an integer in two different bases, J. Reine Angew. Math )