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!!

  • 1
    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

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