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

In this paper, we shall study the stellar work of Norwegian mathematician Selberg and Hungarian mathematician Erdős in providing an Elementary proof of the well-known $\mathit{\text{Prime}}\mathit{\text{Number}}\mathit{\text{Theorem}}$. In addition to introducing ourselves to the notion of $\mathit{\text{Arithmetic}}\mathit{\text{Functions}}$, we shall primarily focus our research on obtaining suitable estimates for the $\mathit{\text{Chebyshev}}\mathit{\text{Theta}}\mathit{\text{Function}}$ $\vartheta (x)$. 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.