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

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