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

TL;DR

This paper introduces and analyzes the recursive divisor function ppa_x(n), deriving its Dirichlet series, identities, and its relation to ordered factorizations, expanding understanding of divisor-related arithmetic functions.

Contribution

It defines the recursive divisor function ppa_x(n), computes its Dirichlet series, and establishes new identities and relations to ordered factorizations, advancing divisor function theory.

Findings

Dirichlet series of ppa_x(n) is ta(s-x)/(2 - ta(s))

ppa_x * ta_y = ppa_y * ta_x, relating ppa_x and ta_y

ppa_0 equals the convolution of the constant function and the number of ordered factorizations K(n)

Abstract

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, $\kappa_0 = {\bf 1} * K$.