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

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.