Type: Article

Digit sums of binomial sums

Arnold Knopfmacher; Florian Luca;

Abstract:

Let b >= 2 be a fixed positive integer and let S(n) be a certain type of binomial sum. In this paper, we show that for most n the sum of the digits of S(n) in base b is at least c(0) logn/(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 coefficients (2n n) and Apery numbers A(n). The proof uses a result of McIntosh regarding the asymptotic expansions of such binomial sums as well as Baker's theorem on lower bounds for nonzero linear forms in logarithms of algebraic numbers.