On the distribution of the truncated sum-of-digits function of polynomial sequences in residue classes

Abstract

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.