  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.
