Generalizations of the ErdH{o}s-Kac Theorem and the Prime Number Theorem

In this paper, we study the linear independence between the distribution of the number of prime factors of integers and that of the largest prime factors of integers. Respectively, under a restriction on the largest prime factors of integers, we will refine the Erd\H{o}s-Kac Theorem and Loyd's recent result on Bergelson and Richter's dynamical generalizations of the Prime Number Theorem. At the end, we will show that the analogue of these results holds with respect to the Erd\H{o}s-Pomerance Theorem as well.