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In this paper, we shall study the stellar work of Norwegian mathematicianSelberg and Hungarian mathematician Erd\H{o}s in providing an Elementary proofof the well-known \textit{Prime Number Theorem}. In addition to introducingourselves to the notion of \textit{Arithmetic Functions}, we shall primarilyfocus our research on obtaining suitable estimates for the \textit{ChebyshevTheta Function} $\vartheta(x)$. Furthermore, we'll try to infer about theasymptotic properties of another function $\rho(x)$, which shall be neededlater on in establishing an equivalent statement of our main result. All themathematical terminologies pertinent to the proof have been discussed in theearlier sections of the text.

On the proof of the Prime Number Theorem using Order Estimates for the Chebyshev Theta Function