Question

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?

Answer 1

This question is known as the multiplication table problem, and was originally posed by Erdos in 1955. Erdos proved that $|A_n|=o(n^2)$, and this was sharpened by Tenenbaum in 1984. In 2008, Ford gave the exact magnitude and proved that $$\left|\{a\cdot b:\ a,b\leq N\}\right|\asymp \frac{N^2}{(\log N)^c(\log\log N)^{3/2}}$$ where $$c=1-\frac{(1+\log \log 2)}{\log 2}.$$

In 2010 Koukoulopoulos gave multidimensional generalizations of Ford's result, proving that $$\left|\{a_1\cdots a_k\ :\ a_i\leq N \text{ for all } \ i\}\right|\asymp \frac{N^{k+1}}{(\log N)^{c_k}(\log\log N)^{3/2}}$$ where $$c_{k}=\int_{1}^{\frac{k}{\log(k+1)}}\log x\text{d}x=\frac{\log(k+1)+k\log\left(k\right)-k\log\log(k+1)-k}{\log(k+1)}.$$

Some references:

• Ford 2008: The distribution of integers with a divisor in a given interval. arXiv link

• Koukoulopoulos 2010: Localized Factorization of Integers. arXiv link

• Koukoulopoulos 2012: On the number of integers in a generalized multiplication table. arXiv link

Remark: The dates used above refer to the publication dates. (Not necessarily the date posted to the arXiv)

Answer 2

The answer is yes, for further infos see the references given at the On-Line Encyclopedia of Integer Sequences.