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

TL;DR

The article presents an analytic route to the Prime Number Theorem by recasting prime-distribution statements in terms of the second Chebyshev function $\psi(x)$ and its contour-integral representations involving the Riemann zeta function $\zeta(s)$. It derives a contour integral for $\frac{\psi_1(x)}{x^2}$ that involves the logarithmic derivative $-\frac{\zeta'(s)}{\zeta(s)}$, and establishes robust upper bounds for $|\zeta(s)|$ and $|\zeta'(s)|$ near the line $\sigma=1$, along with the crucial non-vanishing of $\zeta(s)$ on $\sigma=1$. By shifting the integration contour to $\sigma=1$ and employing the Riemann-Lebesgue lemma, the paper shows $\psi_1(x)/x^2\to 1/2$ and therefore $\psi(x)\sim x$, which is equivalent to the PNT via $\pi(x)\sim x/\log x$. The approach hinges on analytic continuation, Euler product representations, and the functional equation of $\zeta(s)$ to control the growth of $\zeta(s)$ and its logarithmic derivative, yielding a self-contained analytic proof with explicit contour arguments.

Abstract

This article provides a proof of the famous \textit{Prime Number Theorem} by establishing an analogous statement of the same in terms of the second \textit{Chebyshev Function} $ψ(x)$. We shall be extensively using complex analytic techniques in addition to certain meromorphic properties of the \textit{Reimann Zeta Function} $ζ(s)$ and its \textit{Analytic Continuation Property} a priori using Riemann's Functional Equation in order to establish our desired result.

Figures (1)

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Theorems & Definitions (65)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Theorem 2.8
• Theorem 2.9
• proof