SMOOTHING SMOOTH NUMBERS

An integer is called y-smooth if all of its prime factors are less-tha n-or-equal-to y. An important problem is to show that the y-smooth int egers up to x are equi-distributed among short intervals. In particula r, for many applications we would like to know that if y is an arbitra rily small, fixed power of x then all intervals of length square-root x, up to x, contain, asymptotically, the same number of y-smooth integ ers. We come close to this objective by proving that such y-smooth int egers are so equi-distributed in intervals of length square-root x y2epsilon, for any fixed epsilon > 0.