Article
Version 2
Preserved in Portico This version is not peer-reviewed
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)
Version 2 : Received: 20 August 2024 / Approved: 21 August 2024 / Online: 22 August 2024 (02:51:25 CEST)
Version 2 : Received: 20 August 2024 / Approved: 21 August 2024 / Online: 22 August 2024 (02:51:25 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.v2 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.v2
Abstract
This article provides a proof that the Ramanujan's Inequality given by, \begin{align*} \pi(x)^2 < \frac{e x}{\log x} \pi\left(\frac{x}{e}\right) \end{align*} holds unconditionally for every $x\geq \exp(43.5102146)$. 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.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment