Weighted Erdős-Kac theorems via computing moments

By adapting the moment method developed by Granville and Soundararajan (2007), Khan, Milinovich and Subedi (2022) obtained a weighted version of the Erd & odblac;s- Kac theorem for omega(n) with multiplicative weight dk(n), where omega(n) denotes the number of distinct prime divisors of a positive integer n, and dk(n) is the k-fold divisor function with k is an element of N. In the present paper, we generalize their method to study the distribution of additive functions f(n) weighted by nonnegative multiplicative functions alpha(n) in a wide class. In particular, we establish uniform asymptotic formulas for the moments of f(n) with suitable growth rates. We also prove a qualitative result on the moments which extends a theorem of Delange and Halberstam (1957). As a consequence, we obtain a weighted analogue of the Kubilius-Shapiro theorem.