Recursive divisor properties

We recently introduced a new arithmetic function called the recursive divisor function, $\kappa_x(n)$. Here we show that its Dirichlet series is $\zeta(s - x)/(2 - \zeta(s))$, where $\zeta(sigma)$ is the Riemann-zeta function. We relate the recursive divisor function to the ordinary divisor function, $\kappa_0 \times \sigma_x = \kappa_x \times \sigma_0$, which yields a number of new Dirichlet convolutions. We also state the equivalent of the Riemann hypothesis in terms of $\kappa_x$.