By Dirchlet's hyperbola method, one can prove that the average number of divisors of integers $1 \leq n \leq X$ is $\log X$. This question concerns the number of integers $n \leq X$ such that the number of divisors, $d(n)$, is substantially larger than average. Indeed, what is known about the size of the set

$$\displaystyle \{1 \leq n \leq X : d(n) > (\log X)^A \}$$

where $A > 1$ is considered to be a large (but fixed) positive number?