Let $r\in (0,1)$ and denote by $A_r$ the set $A_r=\left\{ a,b\in\mathbb{N} ~:~ a,b\leq N, a\cdot b\leq rN^2 \right\}$. Is it possible to find a good estimation for $|A_r|$?

It is known that $|A_r|= r(1-\log r)N^2+O_r(N)$ where $O_r(\cdot )$ means that it is a function of $r$, but is it possible to say something more accurate about the rate of convergence. To be more specific, what can we say if we take $r=\frac{1}{N}$?

Thanks!!