prime counting function

The prime counting function is a non-multiplicative function for any positive real number $x$, denoted as $\pi (x)$ and gives the number of primes not exceeding $x$. It usually takes a positive integer $n$ for an argument. The first few values of $\pi (n)$ for $n=1,2,3,\mathrm{\dots }$ are $0,1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8\mathrm{\dots }$ (http://www.research.att.com/ njas/sequences/eisA.cgi?Anum=000720OEIS A000720).

The asymptotic behavior of $\pi (x)\sim x/\ln x$ is given by the prime number theorem. This function is closely related with Chebyshev’s functions $\vartheta (x)$ and $\psi (x)$.