For ∅ ̸= P ⊆ P, let DP be the arithmetic subderivative function with respect to P on Z+, let ζDP be the function defined by the Dirichlet series of DP , and let σDP denote its abscissa of convergence. Under certain assumptions concerning s and P, we present asymptotic formulas for the partial sums of ζDP (s) and show that σDP = 2. We also express ζDP (s), s > 2, using the Riemann zeta function.