  We recently introduced the recursive divisor function $\kappa_x(n)$, a
recursive analogue of the usual divisor function. Here we calculate its
Dirichlet series, which is ${\zeta(s-x)}/(2 - \zeta(s))$. We show that
$\kappa_x(n)$ is related to the ordinary divisor function by $\kappa_x *
\sigma_y = \kappa_y * \sigma_x$, where * denotes the Dirichlet convolution.
Using this, we derive several identities relating $\kappa_x$ and some standard
arithmetic functions. We also clarify the relation between $\kappa_0$ and the
much-studied number of ordered factorizations $K(n)$, namely, $\kappa_0 = {\bf
1} * K$.
