# Density of integers with many divisors

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?

• Karl K. Norton, "On the frequencies of large values of divisor functions", Acta Arithmetica 68:3 (1994), 219-244. See eudml.org/doc/206657 Sep 20 at 21:28

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 $$A>\log 2$$, $$|\{1\le n\le X:d(n) \ge (\log X)^A\}| \asymp_A \frac{X}{(\log X)^{B(A)} (\log\log X)^{1/2}}.$$
The normal order of $$\log(d(n))$$ is $$\log(2)\log\log(n))$$. So, for every $$\epsilon>0$$, $$\log(d(n))<(1+\epsilon)\log(2)\log(\log(n))$$ hold for almost all n: that is, if the proportion of $$n\le x$$ for which this does not hold tends to 0 as $$x$$ tends to infinity. Thus $$d(n) < \log(n)^{(1+\epsilon)\log(2)}$$ holds for almost all $$n$$.
• Is it possible to give an explicit bound for the number of such $n$? Sep 20 at 21:11
• Sorry, I meant the density of such numbers as a function of $X$. For example can one obtain a bound of $O_A(X(\log X)^{-B(A)})$ for some number $B(A)$ depending on $A$? Sep 20 at 21:18