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

In this article, our primary objective is to provide an extensive introduction to the Riemann Zeta Function $\zeta(s)$, an integral part in the study different Analytic aspects relevant to the proof of the famous "Prime Number Theorem" (PNT). The whole paper comprises of three mutually exclusive parts. In the initial sections, we define all the necessary terminologies and results handpicked from the areas related to Analytic Number Theory and Analysis of Complex Numbers, whereas, the second half provides an exposition on meromorphic properties of $\zeta(s)$, with brief deduction of the Riemann's Functional Equation for the same. The final and most important part of this essay is solely dedicated towards appreciating the significance of the proof of the Prime Number Theorem, if at all observed from an Analytic point of view. Substantial amount of references in any of the topics pertinent to the contents of this paper have been included in the \texttt{References} section for the reader's convenience, which I do believe shall prove to be resourceful to its ardent readers and scholars.Comment: 26 Page