Exact Form of Prime-Counting Function

Theorem
Let:
 * $\displaystyle \Pi \left({x}\right) = \text{Li} \left({x}\right) - \sum_\rho \text{Li} \left({x^\rho}\right) - \ln \left({2}\right) + \int_x^\infty \frac{\mathrm d t} {t\left({t^2 - 1}\right) \ln \left({t}\right)}$

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

Then the prime-counting function is precisely:


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

where $\mu \left({n}\right)$ denotes the Möbius function.