Prime counting function

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Prime counting function is defined as a function which gives the number of primes before a particular number. It is denoted by \( \pi(x)\).

It was conjectured in the end of the 18\(^\text{th}\) century by Gauss and by Legendre to be approximately \(\displaystyle \pi(x)=\frac{x}{\ln x} \) in the sense that \( \lim_{x \to \infty} \frac{\pi(x)}{x/\ln x}=1 .\)

However, more precise estimates of prime counting function would be

\[\pi(x)=li(x)+O\left(x{e}^{-\sqrt{\ln x}/15}\right).\]

where,\(li(x)=\)logarithmic integral and \(O\) is the big O notation

Example Question 1

\(\pi(4)=2\)