On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $π(x)$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917762160394240 |
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| author | De, Subham |
| author_facet | De, Subham |
| contents | This article provides a proof that the Ramanujan's Inequality given by, $$π(x)^2 < \frac{e x}{\log x} π\Big(\frac{x}{e}\Big)$$ holds unconditionally for every $x\geq \exp(43.5102147)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $π(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02591 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $π(x)$ De, Subham Number Theory Primary 11A41, 11A25, 11N05, 11N37 Secondary 11Y99 This article provides a proof that the Ramanujan's Inequality given by, $$π(x)^2 < \frac{e x}{\log x} π\Big(\frac{x}{e}\Big)$$ holds unconditionally for every $x\geq \exp(43.5102147)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $π(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally. |
| title | On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $π(x)$ |
| topic | Number Theory Primary 11A41, 11A25, 11N05, 11N37 Secondary 11Y99 |
| url | https://arxiv.org/abs/2408.02591 |