# Prime counting function

###### This wiki is incomplete.

For further help in starting a wiki page, check out Wiki Guidelines and Wiki Formatting or come chat with us.

## Definition

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

## \(\pi(4)=2\)

**Cite as:**Prime counting function.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/prime-counting-function/