Digit sums of binomial sums

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 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 coefficients ( 2 n n ) and Apéry numbers A n . The proof uses a result of McIntosh regarding the asymptotic expansions of such binomial sums as well as Bakerʼs theorem on lower bounds for nonzero linear forms in logarithms of algebraic numbers.