The root function γ(n) is
defined by Golomb for n > 1 as the number of distinct representations n = ab with
positive integers a and b. In this paper we define a convolution ∇ such that γ is the
∇-analog of the (Dirichlet) divisor function τ. The structure of the ring of arithmetic
functions under addition and ∇ is discussed. We compute and interpret
∇-analogs of the Moebius function and Euler’s Φ-function. Formulas and an
algorithm for computing the number of distinct representations of an integer
n ≧ 2 in the form n = a1a2⋅⋅cdotak, with ai a positive integer, i = 1,⋯,k, are
given.
