Weighted Erdős-Kac Theorems via Computing Moments

2023/06/20 by Fan, Steve
#11K65 #11N37 #11N60 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.11289

Abstract

By adapting the moment method developed by Granville and Soundararajan [17], Khan, Milinovich and Subedi [24] recently obtained a weighted version of the Erdős--Kac theorem for ω(n) with multiplicative weight d_k(n), where ω(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∈ℕ. In this paper, we generalize their method to study the distribution of additive functions f(n) weighted by nonnegative multiplicative functions α(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.

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