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The '''prime counting function''', denoted <math>\pi</math>, is a [[function]] defined on [[real number]]s.  The quantity <math>\pi(x)</math> is defined as the number of [[positive]] [[prime number]]s less than or equal to <math>x</math>. Gauss first conjectured that the [[prime number theorem]] <math>\sim</math> <math>\frac{x}{log x}</math>, or equivalently, <math>\lim_{x\to\infty}\frac{\pi(x)}{\frac{x}{log x}} = 1</math>.

The function <math>\pi(x)</math> is [[asymptotically equivalent]] to <math>\frac{x}{log x}</math>.  This is the [[prime number theorem]].  It is also asymptotically equivalent to [[Chebyshev's theta function]]. It was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin, working independently. 

The prime counting function has many ties to the [[Riemann zeta function]] and other branches of [[analytic number theory]]. For example, showing that  <math>\pi(x) -  \li(x) =</math> <math>\mathcal{0}(sqrt(x) log x)</math>.

== See also ==

* [[Prime Number Theorem]]
*[[Riemann zeta function]]

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[[Category:Number theory]]
[[Category:Analytic number theory]]