# Recursive* *divisor properties

Number theory

The recursive divisor function is found to have a simple generating function, which leads to a number of new Dirichlet convolutions.

## Properties of the recursive divisor function

Draft (2023)

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 $\kappa_x$.

Draft (2023)