# 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 $n=1,2,3,\ldots$ are $0,1,2,2,3,3,4,4,4,4,5,5,6,6,6,6,7,7,8,8\ldots$ (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 $\psi(x)$.

Title prime counting function PrimeCountingFunction 2013-03-22 12:49:00 2013-03-22 12:49:00 XJamRastafire (349) XJamRastafire (349) 13 XJamRastafire (349) Definition msc 11A25 msc 11A41 msc 11N05 LogarithmicIntegral2