Weighted Erd\H{o}s-Kac Theorems via Computing Moments

By adapting the moment method developed by Granville and Soundararajan [17], Khan, Milinovich and Subedi [24] recently obtained a weighted version of the Erd\H{o}s--Kac theorem for $\omega(n)$ with multiplicative weight $d_k(n)$, where $\omega(n)$ denotes the number of distinct prime divisors of a positive integer $n$, and $d_k(n)$ is the $k$-fold divisor function with $k\in\mathbb{N}$. In this 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 [8]. As a consequence, we obtain a weighted analogue of the Kubilius--Shapiro theorem.