Improved estimate for the prime counting function $\pi(x)$

TL;DR

This paper introduces a new estimate for the prime counting function (x) using combinatorial arguments, improving upon previous bounds by refining the error term and providing a more precise asymptotic approximation.

Contribution

It presents a novel estimate for (x) that improves the error bounds compared to classical results, using combinatorial techniques.

Findings

(x) = (x) + O(1/\u00a0log x)

(x) is approximated by (x) with a smaller error term

The new estimate refines previous asymptotic bounds for (x)

Abstract

Using some simple combinatorial arguments, we establish some new estimates for the prime counting function and its allied functions. In particular we show that \begin{align}\pi(x)=\Theta(x)+O\bigg(\frac{1}{\log x}\bigg), \nonumber \end{align}where \begin{align}\Theta(x)=\frac{\theta(x)}{\log x}+\frac{x}{2\log x}-\frac{1}{4}-\frac{\log 2}{\log x}\sum \limits_{\substack{n\leq x\\\Omega(n)=k\\k\geq 2\\2\not| n}} \frac{\log (\frac{x}{n})}{\log 2}.\nonumber \end{align}This is an improvement to the estimate \begin{align}\pi(x)=\frac{\theta(x)}{\log x}+O\bigg(\frac{x}{\log^2 x}\bigg)\nonumber \end{align}found in the literature.