# How many pairs of integer numbers with bounded product?

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!!

• For $r = 1/N$, this is exactly Dirichlet's divisor problem, cf. the corresponding section of en.wikipedia.org/wiki/…. The next term is of the form $cN + O(N^{\theta})$, and the best value for $\theta$ is still unknown (conjectured to be $1/4$). – js21 Dec 6 at 15:37
• It's not clear from the notation what exactly you are counting. Are the elements of $A_r$ the ordered pairs $(a,b)$? – Gerry Myerson Dec 6 at 21:06