The number of factorizations of numbers less than $x$ into divisors greater than $y$

Abstract:

Let $A(x, y)$ be the number in the title. There is a function $h:[0, \infty ) \to [0, 2]$, decreasing and convex, with $h(0) = 2$ and $\operatorname {lim}_{r \to \infty }h(r) = 0$, such that if $r = \operatorname {log} y/\sqrt {\operatorname {log} x}$ then as $x \to \infty$ with $r$ fixed, \[ A(x,y) = \frac {{C(r)x \operatorname {exp} \left ( {h(r)\sqrt {\operatorname {log} x} } \right )}}{{{{(\operatorname {log} x)}^{3/4}}}}\left ( {1 + O{{(\operatorname {log} x)}^{ - 1/4}}} \right )\]. The estimate is uniform on intervals $0 < r \leq {R_0}$. As corollaries we have for $\operatorname {log} y = \theta {(\operatorname {log} x)^{1/4}}$, \[ \lim \limits _{x \to \infty } \frac {{A(x, y)}} {{A(x, 1)/y}} = {e^{{\theta ^2}/2}}\],and if $\operatorname {log} y = o {(\operatorname {log} x)^{1/4}}$ then $A(x, y) \approx A(x, 1)/y$.