Here's a question I've been asked:
Let $n\in \mathbb{N}$ and let $d_k(n)$ be the number of solutions of $x_1\dots x_k = n, \hspace{5mm}x_i\in \mathbb{N}$
I need to show
$d_k(n) = \sum_{d|n}d_{k-1}(d), \hspace{5mm} k \ge 2$ and that for any $\epsilon > 0$, $d_k(n) = O(n^\epsilon)$ where the implied constant depends only on $k$. I.e $d_k(n) \ll f(k)n^\epsilon$ for some function $f$.
I have to admit, I'm lacking an idea of what $d_k(n)$ looks like. Setting up the base case for an inductive argument isn't difficult, but I'm having a hard time picturing how this being true for an arbitrary $k$ implies the statement for $k+1$. The statement $\epsilon > 0$, $d_k(n) = O(n^\epsilon)$ is also very strange to me, having a CS background. I can kind of see how the choice of $\epsilon$ is arbitrary, since it just varies the implied constant in $O(n^\epsilon$). I can't seem to reconcile any of this in my head. Advice? Thanks for your time.