# Prime counting function

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## 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/