On Ramanujan's prime counting inequality

TL;DR

This paper establishes a new upper bound for the smallest integer beyond which Ramanujan's prime counting inequality always holds, advancing understanding of prime distribution bounds.

Contribution

It provides a novel, tighter upper bound for the critical integer in Ramanujan's prime counting inequality, improving previous estimates.

Findings

New upper bound for N_R established

Inequality holds for all x ≥ N_R with improved bounds

Advances understanding of prime counting function behavior

Abstract

In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $\pi(x)$ holds for every $x \geq N_{\mathcal{R}}$.