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. 



Proof in the literature of an equality involving the prime counting function
Stopple, A Primer of Analytic Number Theory, proves a theorem which looks something like the one under discussion. On page 248, he has $$\pi(x)=R(x)+\sum\_{\rho}R(x^{\rho})+\sum\_{n=1}^{\infty}{\mu(n)\over n}\int\_{x^{1/n}}^{\infty}{dt\over t(t^2-1)\log t}$$


You say that the literature treats your formula as a fact, but you give no citation. Where in the literature do you find your formula? 


