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

TL;DR

This paper explores elementary methods to prove the Prime Number Theorem by estimating the Chebyshev Theta Function and analyzing related arithmetic functions, building on classical work by Selberg and Erdős.

Contribution

It introduces new estimates for the Chebyshev Theta Function and investigates the asymptotic behavior of related functions to support an elementary proof of the Prime Number Theorem.

Findings

Derived suitable bounds for the Chebyshev Theta Function

Analyzed asymptotic properties of the function ρ(x)

Provided insights into elementary proof techniques for prime distribution

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.