Distribution of exponential functions with squarefull exponent in residue rings

Abstract

For a real x ≥ 1 we denote by S[ x] the set of squarefull integers n ≤ x, that is, the set of positive integers n ≤ such that l 2| n for any prime divisor l| n. We estimate exponential sums of the form T a (m,x) = ∑ n∈S[x] exp(2πiaϑ n/m) where ν is a fixed integer with gcd( ν, m) = 1, and apply them to studying the distribution of the powers ν n , n ϵ S[ x], in the residue ring modulo m ≥ 1.