On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $π(x)$
Year: 2024
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: De, Subham
Institution: Indian Institute of Technology Delhi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4406490445