Character sums with exponential functions over smooth numbers

ABSTRACT We give nontrivial bounds in various ranges for exponential sums of the form n∈S (x,y) exp(2πiaϑ n /m) and 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 S t (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