The Prime Number Theorem is one of the most famous theorem in mathematics. It states that the number of primes not exceeding n is asymptotic to 
, where log(n) is the logarithm of (n) to the base e.
The number of primes not exceeding n is commonly written as 
, and an asymptotic relationship between a(n) and b(n) is commonly designated as a(n)~b(n). (This does not mean that a(n)-b(n) is small as n increases. It means the ratio of a(n) to b(n) approaches one as n increases.)
The Prime Number Theorem thus states that ~
.
In other words, the limit (as n approaches infinity) of the ratio of pi(n) to n/log(n) is one. Put a third way, n/log(n) is a good approximation for .
Equivalent Statements
Carl Friedrich Gauss conjectured the equivalent statement that 
was asymptotic to 
defined as:
.
In fact, for large x this turns out to be a better approximation than .