Prime counting function

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

$\pi(n)\sim\frac{\ln(n)}{n}$ .

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

$\pi(x)=\sum_{n}\frac{\mu(n)}{n}J(\sqrt[n]{x})$ ,

where,

μ(n)

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

ln(x)

x

$Li(x)=\int_{0}^{x}\frac{1}{\ln{t}}dt$

ρ

are the non-trivial zeros of the Riemann Zeta function.

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

The convergence of

\sum L i (x ρ) ρ

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

∑	Li(x^ρ)
ρ