Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for smooth numbers

We prove Bombieri--Vinogradov and Barban--Davenport--Halberstam type theorems for the y-smooth numbers less than x, on the range log^{K}x \leq y \leq x. This improves on the range \exp{log^{2/3 + \epsilon}x} \leq y \leq x that was previously available. Our proofs combine zero-density methods with direct applications of the large sieve, which seems to be an essential feature and allows us to cope with the sparseness of the smooth numbers. We also obtain improved individual (i.e. not averaged) estimates for character sums over smooth numbers.