Prime Number Theorem

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 ${\frac {n}{\log(n)}}$, where log(n) is the logarithm of (n) to the base e.

The number of primes not exceeding n is commonly written as $\pi (n)$ the Prime counting function, 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 $\pi (n)$~$n/\log(n)$ .

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 $\pi (n)$.

Equivalent Statements

Carl Friedrich Gauss conjectured the equivalent statement that $\pi (x)$ was asymptotic to ${\mbox{Li}}(x)$ defined as:

${\mbox{Li}}(x)=\int _{2}^{x}{\frac {dt}{\ln t}}$.

In fact, for large x this turns out to be a better approximation than $\pi (x)$. The size of the error ${\mbox{Li}}(x)-\pi (x)$ is closely related to the behavior of the Riemann Zeta function