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If you fix $m$, this is known as the $m$-dimensional multiplication problem. In 2010 Koukoulopoulos showed that as $N\rightarrow \infty$ $$P(m,N)=\left|\lbrace a_1\cdots a_m\ :\ a_i\leq N \text{ for all } \ i\rbrace\right|\asymp \frac{N^{m+1}}{(\log N)^{c_m}(\log\log N)^{3/2}}$$ where $$c_{m}=\int_{1}^{\frac{k}{\log(m+1)}}\log x\text{d}x=\frac{\log(m+1)+m\log\left(m\right)-m\log\log(m+1)-m}{\log(m+1)}.$$ See this answer for more details.