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arXiv · 2308.16303

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

Subham De
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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.

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Subham De. 2024-09-06. On the Order Estimates for Specific Functions of $ζ(s)$ and its Contribution towards the Analytic Proof of The Prime Number Theorem. https://doi.org/10.51483/ijpamr.5.2.2025.35-55

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