In the classic text referred to in the title of this question, the bound $$ H(x,y,2y) \ll \frac{x}{(\log y)^{\delta}\sqrt{\log \log y}},\quad (3\leq y\leq \sqrt{x}) $$ is given, where $\delta=1-\frac{1+\log \log 2}{\log 2}\approx 0.08607\ldots $ after defining $$H(x,y,z)\stackrel{\mathrm{def}}{=} \#\{n\leq x:\text{$n$ has a divisor $d \in (y,z]$}\}.$$ Have there been any improvements on this bound?

Kevin Ford has determined the order of $H(x,y,z)$ for all ranges of $x$, $y$, $z$. In particular, from his work (see Corollary 2) it follows that for $10 \le y\le \sqrt{x}$ one has $$ H(x,y,2y) \asymp \frac{x}{(\log y)^{\delta} (\log \log y)^{3/2}}. $$ In addition to the Annals paper referenced above, you could look at his simpler exposition which deals specifically with $H(x,y,2y)$.

MO 108912$\endgroup$