On the Order Estimates for Specific Functions of $\zeta(s)$ and its Contribution towards the Analytic Proof of The Prime Number Theorem

Abstract

This article provides a proof of the famous \textit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second \textit{Chebyshev Function} $\psi(x)$. We shall be extensively using complex analytic techniques in addition to certain meromorphic properties of the \textit{Reimann Zeta Function} $\zeta(s)$ and its \textit{Analytic Continuation Property} a priori using Riemann's Functional Equation in order to establish our desired result.