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?



Density of integers with many divisors
Theorem 1.11 and Theorem 1.22 of the paper by Norton, cited in the comment of Peter Humphries, show that for any fixed $A \ge \log 2$,
$$
\frac{X (\log\log X)^{O(1)}}{(\log X)^{B(A)}} \ll\_A 
|\{1\le n\le X:d(n) \ge (\log X)^A\}| \ll\_A \frac{X}{(\log X)^{B(A)}},
$$
where
$$
B(A):=1+\frac{A}{\log 2}\left(\log\left(\frac{A}{\log 2}\right) -1 \right).
$$
Equation (1.37) of the same paper gives the correct order of magnitude: For every fixed $A>\log 2$,
$$ 
|\{1\le n\le X:d(n) \ge (\log X)^A\}| \asymp\_A \frac{X}{(\log X)^{B(A)} (\log\log X)^{1/2}}.
$$


