Properties of the recursive divisor function and the number of ordered factorizations

Abstract

We recently introduced the recursive divisor function $κ_x(n)$, a recursive analogue of the usual divisor function. Here we calculate its Dirichlet series, which is ${ζ(s-x)}/(2 - ζ(s))$. We show that $κ_x(n)$ is related to the ordinary divisor function by $κ_x * σ_y = κ_y * σ_x$, where * denotes the Dirichlet convolution. Using this, we derive several identities relating $κ_x$ and some standard arithmetic functions. We also clarify the relation between $κ_0$ and the much-studied number of ordered factorizations $K(n)$, namely, $κ_0 = {\bf 1} * K$.