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

Abstract

In this paper, we shall study the stellar work of Norwegian mathematician Selberg and Hungarian mathematician Erd\H{o}s in providing an Elementary proof of the well-known \textit{Prime Number Theorem}. In addition to introducing ourselves to the notion of \textit{Arithmetic Functions}, we shall primarily focus our research on obtaining suitable estimates for the \textit{Chebyshev Theta Function} $\vartheta(x)$. Furthermore, we'll try to infer about the asymptotic properties of another function $\rho(x)$, which shall be needed later on in establishing an equivalent statement of our main result. All the mathematical terminologies pertinent to the proof have been discussed in the earlier sections of the text.