Prime counting function

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

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

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

<math>\pi(x)=\sum_{n}\frac{\mu(n)}{n}J(\sqrt[n]{x})</math>,

where,

<math>\mu(n)</math> is Möbius Mu function,

<math>J(x)=Li(x)-\sum_{\rho}Li(x^{\rho})-\ln(2)+\int^{\infty}_{x}\frac{dt}{t(t^2-1)\ln(t)}</math>

<math>\ln(x)</math> is the natural logarithm of <math>x</math>

<math>Li(x)=\int_{0}^{x}\frac{1}{\ln{t}}dt</math>

<math>\rho</math> are the non-trivial zeros of the Riemann Zeta function.

Whilst the sum is over all <math>n</math> it is needed only to add up to the term such that <math>\sqrt[n]{x}\leq2</math> as after that <math>J(\sqrt[n]{x})=0</math>.

The convergence of

<math>\sum_{\rho}Li(x^{\rho})</math>

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