Is there any easy way to show $$\lim_{N \to \infty} \frac{1}{N^2}\#\{ab : 1 \le a,b \le N\} = 0$$ A quick calculation I did shows that the number of positive integers $\le N^2$ with a prime divisor $p > N$ is at most the order of $(\log 2) \cdot N^2$, so just getting rid of the numbers with a high prime divisor is not sufficient.
2026-04-21 20:38:34.1776803914
Density of multiplication table
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