The Prime Number Theorem and Primorial Numbers

TL;DR

This paper explores expressing the prime number theorem through primorial numbers and their totatives, offering new functions to approximate prime distribution asymptotically.

Contribution

It introduces a novel approach using primorial numbers and totatives to analyze prime distribution, extending classical prime number theorem concepts.

Findings

Derived new asymptotic functions for prime counting

Connected primorial properties with prime distribution

Enhanced understanding of prime number behavior

Abstract

Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number theorem determines that such asymptotic behavior is similar to the asymptotic behavior of the number divided by its natural logarithm. In this paper, we take advantage of a multiplicative representation of a number and the properties of the logarithm function to express the prime number theorem in terms of primorial numbers, and n-primorial totative numbers. A primorial number is the multiplication of the first n prime numbers while n-primorial totatives are the numbers that are coprime to the n-th primorial number. By doing this we can define several different functions that can be used to approximate the behavior of the prime counting function asymptotically.