On the Summatory Function of <img src=image/13443653_24.gif>

An asymptotic formula is derived for the summatory function <img src=image/13443653_02.gif>, where <img src=image/13443653_03.gif> denotes the total number of prime factors of <img src=image/13443653_16.gif> counted with multiplicity, and <img src=image/13443653_04.gif> is a multiplicative arithmetical function satisfying <img src=image/13443653_05.gif> for primes <img src=image/13443653_06.gif> and non-negative integers <img src=image/13443653_07.gif>, where <img src=image/13443653_08.gif> and <img src=image/13443653_01.gif> for <img src=image/13443653_09.gif>. The study builds on a rich history in analytic number theory, including classical results by Dirichlet on the divisor function <img src=image/13443653_10.gif>, and refinements using zeta-function estimates, as well as probabilistic approaches like the Erdős–Kac theorem extended to <img src=image/13443653_03.gif> (distinct primes) over <img src=image/13443653_11.gif>-free and <img src=image/13443653_11.gif>-full numbers. However, prior research has largely overlooked the multiplicity in <img src=image/13443653_03.gif> and its twisting by broad classes of multiplicative functions beyond divisors, particularly for square-full integers. The analysis covers three distinct cases: when <img src=image/13443653_04.gif> belongs to the subclass <img src=image/13443653_12.gif> (where <img src=image/13443653_13.gif> for all primes <img src=image/13443653_06.gif>); when <img src=image/13443653_04.gif> is in the broader class <img src=image/13443653_15.gif> but not in <img src=image/13443653_12.gif>; and when <img src=image/13443653_16.gif> is square-full with <img src=image/13443653_14.gif>. Examples of such functions include the number of non-isomorphic Abelian groups of order <img src=image/13443653_16.gif>, the number of square-full divisors of <img src=image/13443653_16.gif>, the divisor function <img src=image/13443653_17.gif>, and the <img src=image/13443653_07.gif>-fold divisor function <img src=image/13443653_18.gif>. The results are obtained using Dirichlet series <img src=image/13443653_19.gif>, which admit an Euler product decomposition due to multiplicativity, enabling analytic continuation via differentiation with respect to an auxiliary parameter <img src=image/13443653_20.gif>, contour integration, and estimates for the Riemann zeta function, as well as analytic continuation techniques. The following results were obtained: for case (i), <img src=image/13443653_21.gif>; for case (ii), <img src=image/13443653_22.gif>; and for square-full <img src=image/13443653_16.gif>, <img src=image/13443653_23.gif>. 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.