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The '''prime number theorem''' is the name given to several theorems that provide estimates of the number of primes less than or equal to any given number:

The '''Prime Number Theorem''' is one of the most famous theorems in mathematics. It states that the number of primes not exceeding n is [[asymptotic]] to <math>\frac{n}{\log(n)}</math>, where log(n) is the logarithm of (n) to the base e. 

Let π(''n'') be the [[prime counting function]] providing the number of primes less than or equal to ''n'', for any positive number ''n''. The simplest form of the prime number theorem states that

:<math>\pi(n)\sim\frac{n}{\ln n}</math>.

:<math>\pi(n)\sim\frac{n}{\ln n}</math>.

The number of primes not exceeding n is commonly written as <math>\pi(n)</math>, the [[prime counting function]]. 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.)

That is, as n tends to infinity, the [[relative error]] between π(''n'') and ''n''/(ln ''n'') tends to zero. This can be expressed using limit notation as  

The Prime Number Theorem thus states that <math> \pi(n) </math>~<math> n/ \log(n)</math> . That is, as n tends to infinity, the [[relative error]] between <math> \pi(n) </math> and <math> n/ \log(n)</math> tends to zero. This can be expressed using limit notation as  

:<math>\lim_{n\to\infty}\frac{\pi(n)}{n/\ln(n)}=1</math>

:<math>\lim_{n\to\infty}\frac{\pi(n)}{n/\ln(n)}=1</math>

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, <math>n/log(n)</math> is a good approximation for <math>\pi(n)</math>.

 

 

 

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 <math>\pi(n)</math>.

== Equivalent Statements ==

== Equivalent Statements ==