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

Let q ≥ 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 mod q k)) n < q k and (s p (n d mod p k)) n < p k in residue classes modulo m, where q ≥ 2 , m ≥ 2 and d ≥ 2 are general integers, p > 2 is a prime. Furthermore, we give exact identities for the distribution of (s p (n d 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. [ABSTRACT FROM AUTHOR]