Let \(q \geq 2\) be an integer and let \(s_{q}(n)\) be the sum-of-digits function in base q of the positive integer n. In this paper we obtain asymptotic formulae for the distribution of \((s_q(n^2 {\rm mod} q^k))_{n<q^k}\) and \((s_p(n^d {\rm mod} p^k))_{n<p^k}\) in residue classes modulo m, where \(q\geq 2\), \(m \geq 2\) and \(d \geq 2\) are general integers, \(p > 2\) is a prime. Furthermore, we give exact identities for the distribution of \((s_p(n^{d} {\rm mod} p^k))_{n<p^k}\)in residue classes modulo p. The properties of Dirichlet character sums and exponential sums play an important role in the proof of the results.

让\(q \geq 2\)是一个整数，让\(s_{q}(n)\)是正整数n 的基数q中的数字总和函数。在本文中，我们获得了\((s_q(n^2 {\rm mod} q^k))_{n<q^k}\)和\((s_p(n^d {\ rm mod} p^k))_{n<p^k}\)在以m为模的残基类中，其中\(q\geq 2\) , \(m \geq 2\)和\(d \geq 2\ )是一般整数，\(p > 2\)是素数。此外，我们给出了\((s_p(n^{d} {\rm mod} p^k))_{n<p^k}\)在以p 为模的残基类中分布的精确恒等式. Dirichlet 字符和和指数和的性质在结果的证明中起着重要作用。