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On Proving Ramanujan’s Inequality using a Sharper Bound for the Prime Counting Function π(x)
Version 1
: Received: 5 August 2024 / Approved: 6 August 2024 / Online: 8 August 2024 (11:53:54 CEST)
How to cite: De, S. On Proving Ramanujan’s Inequality using a Sharper Bound for the Prime Counting Function π(x). Preprints 2024, 2024080451. https://doi.org/10.20944/preprints202408.0451.v1 De, S. On Proving Ramanujan’s Inequality using a Sharper Bound for the Prime Counting Function π(x). Preprints 2024, 2024080451. https://doi.org/10.20944/preprints202408.0451.v1
Abstract
This article provides a proof that the Ramanujan's Inequality given by, $$\pi(x)^2 < \frac{e x}{\log x} \pi\Big(\frac{x}{e}\Big)$$ holds unconditionally for every $x\geq \exp(59)$. 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 $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.
Keywords
ramanujan; prime counting function; chebyshev theta function; mathematica; primes
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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