in mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. it is denoted by \scriptstyle\pi(x) (this does not refer to the number Ï). thumb|right|400px|the values of Ï(n) for the first 60 integers ==history== of great interest in number theory is the growth rate of the prime-counting function. it was conjectured in the end of the 18th century by gauss and by legendre to be approximately : x/\operatorname{ln}(x)\! in the sense that :\lim_{x\rightarrow\infty}\frac{\pi(x)}{x/\operatorname{ln}(x)}=1.\! this statement is the prime number theorem. an equivalent statement is :\lim_{x\rightarrow\infty}\pi(x) / \operatorname{li}(x)=1\! where li is the logarithmic integral function. the prime number theorem was first proved in 1896 by jacques hadamard and by charles de la vallÃ©e poussin independently, using properties of the riemann zeta function introduced by riemann in 1859. more precise estimates of \pi(x)\! are now known; for example :\pi(x) = \operatorname{li}(x) + o\bigl(xe^{-\sqrt{\ln x}/15}\bigr)\! where the o is big o notation. for most values of x we are interested in (i.e., when x is not unreasonably large) \operatorname{li}(x)\! is greater than \pi(x)\!, but infinitely often the opposite is true. for a discussion of this, see skewes' number. proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by atle selberg and by paul erdÅs (for the most part independently). ==table of Ï(x), x / ln x, and li(x)== the table shows how the three functions Ï(x), x / ln x and li(x) compare at powers of 10. see also,{{cite w
Analytic_number_theory Prime_numbers