On proving an Inequality of Ramanujan using Explicit Order Estimates for the Mertens Function
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866910570870996992 |
|---|---|
| author | De, Subham |
| author_facet | De, Subham |
| contents | This research article provides an unconditional proof of an inequality proposed by Srinivasa Ramanujan involving the Prime Counting Function $π(x)$,
\begin{align*}
(π(x))^{2}<\frac{ex}{\log x}π\left(\frac{x}{e}\right)
\end{align*} for every real $x\geq \exp(547)$, using specific order estimates for the Mertens Function, $M(x)$. The proof primarily hinges upon investigating the underlying relation between $M(x)$ and the Second Chebyshev Function, $ψ(x)$, in addition to applying the meromorphic properties of the Riemann Zeta Function, $ζ(s)$ with an intention of deriving an improved approximation for $π(x)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12052 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On proving an Inequality of Ramanujan using Explicit Order Estimates for the Mertens Function De, Subham General Mathematics Primary 11A41, 11A25, 11N05, 11N37, 11N56 Secondary 11M06, 11M26 This research article provides an unconditional proof of an inequality proposed by Srinivasa Ramanujan involving the Prime Counting Function $π(x)$, \begin{align*} (π(x))^{2}<\frac{ex}{\log x}π\left(\frac{x}{e}\right) \end{align*} for every real $x\geq \exp(547)$, using specific order estimates for the Mertens Function, $M(x)$. The proof primarily hinges upon investigating the underlying relation between $M(x)$ and the Second Chebyshev Function, $ψ(x)$, in addition to applying the meromorphic properties of the Riemann Zeta Function, $ζ(s)$ with an intention of deriving an improved approximation for $π(x)$. |
| title | On proving an Inequality of Ramanujan using Explicit Order Estimates for the Mertens Function |
| topic | General Mathematics Primary 11A41, 11A25, 11N05, 11N37, 11N56 Secondary 11M06, 11M26 |
| url | https://arxiv.org/abs/2407.12052 |