Uniform Estimates for Integers with a Large Smooth Part

Source abstract

We study the distribution of integers with a large smooth part. We obtain a uniform estimate for the number of integers up to $x$ whose $y$-smooth part exceeds a given threshold $z$, in the range $\log z\le y\le z\le x/2$. Our estimate has the expected exponential order governed by the Dickman function and holds uniformly throughout this range. The proof combines classical estimates for smooth and rough numbers with uniform estimates for the local behavior of the smooth-number counting function.