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Claim. Write $H(x,y,z)$ for the number of $n\le x$ with a divisor in $(y,z]$, $H_r(x,y,z)$ for those with exactly $r$ such divisors, and $\varepsilon(y,z)$, $\varepsilon_r(y,z)$ for the limits of $H/x$ and $H_r/x$. Ford's Corollary 2, at $c=2$, gives the first answer asked by Problem 446:

$$\delta(n)=\varepsilon(n,2n)\asymp\frac{1}{(\log n)^{\alpha}(\log\log n)^{3/2}}, \qquad \alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\ldots,$$

the order of magnitude of the density of integers with a divisor in $(n,2n)$, with matching upper and lower bounds; this sharpens Erdős's 1960 estimate $\delta(n)=(\log n)^{-\alpha+o(1)}$ and Tenenbaum's 1984 bounds, which match only up to slowly varying factors. For the second question, Theorem 4 gives $H_1(x,y,z)/H(x,y,z)\asymp\log\log(z/y+10)/\log(z/y+10)$ in a wide range, and Corollary 7 states the consequence for the densities: for every $r\ge1$ and $\lambda>1$,

$$\frac{\varepsilon_r(y,\lambda y)}{\varepsilon(y,\lambda y)}\gg_{r,\lambda}1 .$$

With $r=1$ and $\lambda=2$ this is $\delta_1(n)\gg\delta(n)$, so $\delta_1(n)=o(\delta(n))$ is false; Ford states this as the refutation of his Conjecture 1, which he attributes to Erdős. The intervals are half-open, $(y,2y]$, where the problem writes $(n,2n)$; the integers divisible by $2n$ itself have density $1/(2n)$, far below $\delta(n)$, so the orders are the same. The paper is filed as Ford 2008; the method combines sieve-style reductions of $H$ and $H_r$ to averages of divisor-distribution functions with new bounds for uniform order statistics.

Both parts of the problem are settled by this one result: the growth rate is determined, and the second question is answered no. Ford proves more than the problem asks, $\varepsilon_r(y,z)/\varepsilon(y,z)\to0$ whenever $z/y\to\infty$ and the order of $H_r$ for $r\ge2$ in most ranges.

Acceptance. Refereed: Ann. of Math. (2) 168 (2008), no. 2, 367–433. Reviewed: the site's curator, T. F. Bloom, marks Problem 446 solved and credits Ford for both the growth rate and the disproof. The page's date is the arXiv first posting, 2004-01-18. No formalization is recorded, and this repository has not checked the proof independently.