Character sums with exponential functions over smooth numbers

We give nontrivial bounds in various ranges for exponential sums of the form [equation omitted for formatting reasons] and [equation omitted for formatting reasons] where m ≥ 2, ϑ is an element of order t in the multiplicative group Z*m, gcd(a, m) = 1, S(x, y) is the set of y-smooth integers n ≤ x, and St (x, y) is the subset of S (x, y) consisting of integers that are coprime to t. We obtain sharper bounds in the special case that m = q is a prime number.12 page(s