Preprint Article Version 1 This version is not peer-reviewed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)

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

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