Weighted Erdős-Kac Theorems via Computing Moments
Steve Fan
TL;DR
This work extends the Erdős–Kac paradigm to weighted distributions of additive functions by developing a robust, moment-based framework for nonnegative multiplicative weights $\\alpha(n)$ in a broad class $\\mathcal{M}^{\\ast}$. By proving uniform asymptotics for weighted moments $M(x;m)$ in terms of the variance proxy $B(x)$ and the even Gaussian moments $C_m$, the authors establish Gaussian limiting behavior for $(f(n)-A(x))/\sqrt{B(x)}$ under flexible hypotheses, unifying and extending prior weighted results. The paper also handles generalization to compositions $f(g(n))$ with $g$ a nonconstant polynomial, provides a weighted Kubilius–Shapiro analogue, and yields a weighted Halberstam–Delange theorem with corollaries that apply to notable arithmetic functions (e.g., $d_k(n)$, Ramanujan’s tau). The approach is elementary yet adaptable, enabling distribution results for various weighted additive functions and opening avenues for further applications in sieve theory and arithmetic statistics. Overall, the results significantly broaden the reach of weighted Erdős–Kac-type phenomena and provide a versatile toolkit for Gaussian-distribution conclusions in weighted additive-function settings.
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 $k\in\mathbb{N}$. 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.