On the Summatory Function of

An asymptotic formula is derived for the summatory function , where denotes the total number of prime factors of counted with multiplicity, and is a multiplicative arithmetical function satisfying for primes and non-negative integers , where and for . The study builds on a rich history in analytic number theory, including classical results by Dirichlet on the divisor function , and refinements using zeta-function estimates, as well as probabilistic approaches like the Erdős–Kac theorem extended to (distinct primes) over -free and -full numbers. However, prior research has largely overlooked the multiplicity in and its twisting by broad classes of multiplicative functions beyond divisors, particularly for square-full integers. The analysis covers three distinct cases: when belongs to the subclass (where for all primes ); when is in the broader class but not in ; and when is square-full with . Examples of such functions include the number of non-isomorphic Abelian groups of order , the number of square-full divisors of , the divisor function , and the -fold divisor function . The results are obtained using Dirichlet series , which admit an Euler product decomposition due to multiplicativity, enabling analytic continuation via differentiation with respect to an auxiliary parameter , contour integration, and estimates for the Riemann zeta function, as well as analytic continuation techniques. The following results were obtained: for case (i), ; for case (ii), ; and for square-full , . The work is theoretical in nature. The results of this study can be applied in further research in number theory, group theory, and discrete mathematics, with potential applications in algorithmic number theory (e.g., efficient computation of group orders) and cryptographic protocols relying on prime factorizations.