Discussion on the history of the Prime Number Theorem and bounds of $\psi(x)$

In this talk I will be discussing the history of the Prime Number Theorem following the works of Legendre, Gauss, Riemann, Hadamard, and de la Vallée Poussin, followed by a survey on explicit bounds for $\psi(x)$ beginning with the work of Rosser in 1941. I will go over various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, and Büthe. I will finally briefly discuss my work on the survey of a paper by Büthe and its significance.