Prime counting function

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The Prime counting function counts the number of primes less than or equal to . The prime number theorem says that,


In 1859 Bernhard Riemann presented a paper On the number of primes less than a given number he showed this to be exactly,



is Möbius Mu function,
is the natural logarithm of
are the non-trivial zeros of the Riemann Zeta function.

Whilst the sum is over all it is needed only to add up to the term such that as after that .

The convergence of

is dependent on the Riemann hypothesis and if true is better behaved.