For a real x ≥ 1 we denote by Sk[X] the set of k-full integers n ≤ x, that is, the set of positive integers n ≤ x such that ℓk n for any prime divisor ℓ n. We estimate exponential sums of the form T(a, p, x, k) = ∑n∈Sk[x] exp(2πiaθn/p), where θ is a fixed integer with gcd (θ, p) = 1, and apply them to studying the distribution of the powers θn, n ∈ Sk[x], in the residue ring modulo p ≥ 1.

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Journal Indagationes Mathematicae
Dewar, M. (Michael), Panario, D, & Shparlinski, I.E. (Igor E.). (2004). Distribution of exponential functions with k-full exponent modulo a prime. Indagationes Mathematicae, 15(4), 497–503. doi:10.1016/S0019-3577(04)80014-4