On the mean-value of the powers of digital sums (Q2774097)

Let \(s(n)\) denote the sum of the base \(10\) digits of the positive integer \(n,\) and let \(a\) be any fixed positive number. In this paper, the author gets an asymptotic formula for the sum \(\sum_{n<x} s(n)^a\) and gives the main terms and the coefficients to arbitrary order. The estimate is a generalization of the result of \textit{Yu Xiuyuan} and \textit{Cui Zhen} [Acta Math. Sin. 41, 881-888 (1998; Zbl 1020.11060)], where an asymptotic formula of the same type for the case \(a=k\in \mathbb{N}\) was obtained.