Distribution of exponential functions with squarefull exponent in residue rings

AbstractFor 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 l2|n for any prime divisor l|n. We estimate exponential sums of the form Ta (m,x) = ∑n∈S[x] exp(2πiaϑn/m) where ν is a fixed integer with gcd(ν,m) = 1, and apply them to studying the distribution of the powers νn, n ϵ S[x], in the residue ring modulo m ≥ 1