Exact Form of Prime-Counting Function

Theorem
The prime-counting function is precisely

$$\pi(x) = \sum_{n=1}^\infty \left({ \frac{\mu(n)}{n} \Pi(x^{1/n} ) }\right) \ $$,

$$\Pi(x) = \text{Li}(x) - \sum_\rho \text{Li}(x^\rho) - \log(2) + \int_x^\infty \frac{dt}{t(t^2-1)\log(t)} \ $$

where $$\text{Li}(x) \ $$ is the logarithmic integral and the sum is taken over all $$0<\rho\in\R \ $$ such that the zeta function $$\zeta(\alpha+i\rho) =0 \ $$ for some $$\alpha \in \R\ $$.