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On proving an Inequality of Ramanujan using Explicit Order Estimates of the Mertens Function
Version 1
: Received: 8 July 2024 / Approved: 9 July 2024 / Online: 9 July 2024 (16:22:12 CEST)
How to cite: De, S. On proving an Inequality of Ramanujan using Explicit Order Estimates of the Mertens Function. Preprints 2024, 2024070759. https://doi.org/10.20944/preprints202407.0759.v1 De, S. On proving an Inequality of Ramanujan using Explicit Order Estimates of the Mertens Function. Preprints 2024, 2024070759. https://doi.org/10.20944/preprints202407.0759.v1
Abstract
This research article provides an unconditional proof of an inequality proposed by \textit{Srinivasa Ramanujan} involving the Prime Counting Function $\pi(x)$, \begin{align*} (\pi(x))^{2}<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right) \end{align*} for every real $x\geq \exp(1486)$, using specific order estimates of the \textit{Mertens Function}, $M(x)$. The proof primarily hinges upon investigating the underlying relation between $M(x)$ and the \textit{Second Chebyshev Function}, $\psi(x)$, in addition to applying the meromorphic properties of the \textit{Riemann Zeta Function}, $\zeta(s)$ with an intention of deriving an improved approximation for $\pi(x)$.
Keywords
Riemann Zeta Function; Mertens Function; Chebyshev Function; arithmetic function; error estimates; Perron’s Formula; M¨obius Inversion Formula; Dirichlet Partial Summation Formula
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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