On the sum of digits of the factorial

TL;DR

This paper establishes a new lower bound on the sum of digits of factorials in any base, showing it grows faster than previously known, with implications for number theory and digit sum analysis.

Contribution

It provides an improved lower bound for the digit sum of factorials and least common multiples, refining earlier results by Luca.

Findings

s_b(n!) > C_b log n log log log n for n > e

The bound applies to factorials and least common multiples

The result depends only on the base b

Abstract

Let b > 1 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.