The sum-of-digits function in rings of residue classes

Dartyge and Sárközy introduced the notion of digits in finite fields and studied the properties of polynomial values of \({\mathbb {F}}_q\) with a fixed sum of digits. Swaenepoel provided sharp estimates for the number of elements of special sequences of \({\mathbb {F}}_q\) whose sum of digits is prescribed. In this paper we study the sum-of-digits function in rings of residue classes and give a few asymptotic formulas and exact identities by using estimates for character sums and exponential sums modulo prime powers.