Fix an integer $d \geq 2$ and for every real number $x$ let $M\_d(x)$ be number of $d \times d$ matrices $(a\_{ij})$ satisfying: every $a\_{ij}$ is a positive integer, the product of every row does not exceed $x$, and the product of every column does not exceed $x$.


I'm looking for a good upper bound for $M\_d(x)$ as $x \to +\infty$.


If we forget about the condition on the columns, since it is well known that the number of $d$-tuples $(b\_1, \dots, b\_d)$ of positive integers satisfying $b\_1 \cdots b\_d \leq x$ is $\ll\_d x (\log x)^{d - 1}$ (a generalization of Dirichlet divisor problem), we get the upper bound 
$$M\_d(x) \ll\_d x^d (\log x)^{d(d-1)}.$$


In the special case $d = 2$, we have that $a\_{12}, a\_{21} \leq \min(x / a\_{11}, x / a\_{22})$ and consequently
$$M\_2(x) \leq \sum\_{a\_{11}, a\_{22} \leq x} \min\left(\frac{x}{a\_{11}}, \frac{x}{a\_{22}}\right)^2 \ll \sum\_{a\_{11} \leq a\_{22} \leq x} \left(\frac{x}{a\_{22}}\right)^2 \ll x^2 \log x ,$$
which is a better upper bound than the general one given in the above paragraph. However, I have no idea of how to generalize this trick to $d \geq 3$ (if possible).


Has this problem been studied before? Do you have any idea/suggestion about it?



Number of matrices with bounded products of rows and columns
This problem was considered in passing in the proof of Theorem 4.1 in [Granville and Soundararajan](https://arxiv.org/pdf/math/9903196.pdf), see the argument starting at the bottom of page 17. They show (in your notation) that $M\_d(x)$ is of order $x^d (\log x)^{(d-1)^2}$. You should also look at work of [Harper, Nikeghbali and Radziwill](https://arxiv.org/pdf/1505.01443.pdf) which shows an asymptotic formula for closely related objects (see Theorem 3 there). 


