I have a fair die 6-sided die labeled $1$ to $6$ on each side. Let $X_i$ be the face value of the $i^{th}$ roll. How would one go about finding the probability distribution for the product of the resultant rolls, i.e. for $PD=X_1\cdot X_2 \space$: $$PD=\prod_{i=1}^2X_i$$
I know $X_i$ will be independent and clearly brute force is a valid option here but what about when $n$ is large. I thought about the convolution of the probability generating function, but clearly we will have $P(PD=k)=0$ for some $k$. It would be nice if there was a equivalence of partitions and compositions for products of natural numbers.
Is there any theory I may have not seen that could find this product generally.