Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem

Abstract

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 Erdős-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 Erdős-Pomerance Theorem as well.

arXiv comment20 pages. Accepted by the Communications in Mathematics and Statistics

References in corpus (4)

• Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative semigroup actions
• Beyond the LSD method for the partial sums of multiplicative functions
• Analogues of Alladi's formula over global function fields
• Dynamics on the number of prime divisors for additive arithmetic semigroups

Cited by in corpus (2)

• The prime number theorem over integers of power-free polynomial values
• On averages of completely multiplicative functions over co-prime integer pairs