Let $d_k(n)$ be the $k$-divisor function, i.e., it is a number of ways to represent $n$ as a product of $k$ natural numbers.
I'm interested in the asymptotic behavior (or, at least, in upper bounds) of $d_k(n)$ as $n \to \infty$ in the following two cases:
1) $k$ is fixed;
2) $k \to \infty$ in some monotone way, e.g., as $k = C \log(n)$ for some constant $C \in \left(0, \frac{\log(n)}{\log(2)}\right)$.
I saw quite a lot of works about the behavior and bounds for the sum $\sum_{n \leq x} d_k(n)$. However, I was not able to find results concerning the behavior of $d_k(n)$ by itself. Thus, I'll appreciate even references to the proper literature.