  In 2020, Bergelson and Richter gave a dynamical generalization of the
classical Prime Number Theorem, which has been generalized by Loyd in a
disjoint form with the Erd\H{o}s-Kac Theorem. These generalizations reveal the
rich ergodic properties of the number of prime divisors of integers. In this
article, we show a new generalization of Bergelson and Richter's Theorem in a
disjoint form with the distribution of the largest prime factors of integers.
Then following Bergelson and Richter's techniques, we will show the analogues
of all of these results for the arithmetic semigroups arising from finite
fields as well.
