Dynamics on the number of prime divisors for additive arithmetic semigroups

Abstract

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 Erdős-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.

arXiv comment19 pages. To appear in Finite Fields and Their Applications

References in corpus (5)

• Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
• A new elementary proof of the Prime Number Theorem
• Analogues of Alladi's formula
• Analogues of Alladi's formula over global function fields
• Generalizations of Alladi's formula for arithmetical semigroups

Cited by in corpus (4)

• Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem
• The prime number theorem over integers of power-free polynomial values
• On averages of completely multiplicative functions over co-prime integer pairs
• On covering systems of polynomial rings over finite fields