the number of primes under x,pi(x)?

Question

What is known of the function related to prime numbers pi(x),the number of primes under number x.


i want to know as much as possible of function pi(x) the number of primes lower then x

Answer 1

I was just reading about this yesterday. You can check the wiki article on it: The function:

pi(x) = x / In(x) is the first approximation, but later formulas refined the accuracy. Carl Fredrich Gauss, at the age of only FIFTEEN, devised the Li(x) function:

Li(x) = Integral (1 / In(t)) dt from limits 2 to x.

Answer 2

Refer to a book in analytic number theory. This is a very deep question and if you can put a dent in it, you've got yourself a dissertation!