The Zeta Function, Logarithmic Integrals, and the Prime Counting Function

Abstract

The Prime Number Theorem (PNT) states that the number of primes less than a given value x is asymptotically equal to x/log(x). The PNT was first conjectured by Guass, but it was not proved until over 100 years later (in 1896) by Hadamard and de la Vallee-Poussin. We explore the history of the PNT as well as results related to the prime counting function, the logarithmic integral function, and the zeta function.