Exact Form of Prime-Counting Function

Theorem
The prime-counting function is precisely


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


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

where $\text{Li}(x)$ is the offset 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 $\alpha \in \R$.