# Estimates for $\pi(x)$ for large values of $x$ and Ramanujan's prime counting inequality

## TL;DR

This paper improves explicit estimates for the prime counting function 1(x) using refined Chebyshev 1-functions, and applies these to bound the smallest integer where Ramanujan's inequality always holds.

## Contribution

It introduces new explicit bounds for 1(x) based on refined Chebyshev 1-functions, enhancing previous estimates for large x.

## Key findings

- New explicit estimates for 1(x) for large x
- An improved upper bound for the smallest H_0 satisfying Ramanujan's inequality
- Enhanced understanding of prime distribution for large x

## Abstract

In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $\pi(x)$, which improve the current best estimates for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.