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

2023/07/18 by Thomas Fink, Fink, T. M. A.
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2307.09140

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 κ₀ and the much-studied number of ordered factorizations K(n), namely, κ₀ = \bf 1 * K.

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