The number of factorizations of numbers less than $x$ into factors less than $y$

Abstract:

Let $K(x,y)$ be the number in the title. There is a function $f(r)$, concave and decreasing with $f(0) = 2$ and $f’(0) = 0$ such that if $r = \sqrt {\log x} /\log y$ then as $x \to \infty$ with $r$ fixed, \[ K(x,y) = x \exp \left ({f(r) \sqrt {\log x} + O {{(\log \log x)}^2}} \right )\]. The proof uses a uniform version of Chernoff’s theorem on large deviations from the sample mean of a sum of $N$ independent random variables.