Let \begin{equation} R(x) = \sum_{k=1}^{\infty}\frac{\mu(k)}{k}li(x^{1/k}) \end{equation} where $\mu$ is the Mobius function and \begin{equation} li(x) = \int_0^x \frac{dt}{\log t} \end{equation} Is there a proof in the literature of \begin{equation} \pi(x)=R(x)-\sum_{\rho}R(x^{\rho}) \end{equation} where $\pi$ is prime counting function and the sum is over all complex zeros of $\zeta(s)$. The literature seems to treat it as fact while stating no proof is available - a strange situation.

Thanks in advance.