On the sum of digits of the factorial

Abstract

Abstract. Let b ≥ 2 be an integer and denote by s b (m) the sum of the digits of the positive integer m when is written in base b. We prove that s b (n!) > C b log n log log log n for each integer n > e, where C b is a positive constant depending only on b. This improves of a factor log log log n a previous lower bound for s b (n!) given by Luca. We prove also the same inequality but with n! replaced by the least common multiple of 1, 2, . . . , n.