prime counting function PrimeCountingFunction 2002-06-27 13:35:42 2006-12-01 14:11:44 Definition number theory % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\it 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 $ (\PMlinkexternal{OEIS A000720}{http://www.research.att.com/~njas/sequences/eisA.cgi?Anum=000720} ). The asymptotic behavior of $\pi(x) \sim x/\ln x$ is given by the prime number theorem. This function is closely related with {\it Chebyshev's functions} $\vartheta(x)$ and $\psi(x)$.