Arithmetic Subderivatives Relative to Subsets of Primes - Talukdar, Champak; Saikia, Helen K. (Communications in Mathematics and Applications, 2025)
The arithmetic derivative is a number-theoretic function that behaves analogously to differentiation defined on integers. In this paper, we extend the concept to generalized arithmetic subderivatives with respect to a chosen set of primes. Inspired by the work of P. Haukkanen (Generalized arithmetic subderivative, Notes on Number Theory and Discrete Mathematics 25(2) (2019), 1 – 7), we introduce a family of arithmetic functions that generalize the prime-power exponents in the classical definition. We establish necessary and sufficient conditions under which these generalized subderivatives satisfy analogs of linearity, the Leibniz rule (product rule), and multiplicativity. Several illustrative examples are worked out to demonstrate the computations and properties of these subderivatives.