Let $\pi(x)$ denote the prime counting function and $$\pi_0(x) = \lim_{\epsilon \to 0} \frac{\pi(x+\epsilon)+\pi(x-\epsilon)}{2}.$$
I've seen noted in a few references the explicit formula
$$\pi_0(x) = R(x)-\sum_\rho R(x^\rho), \quad x>1,$$
where the sum is over all zeros $\rho$ of the Riemann zeta function (the nontrivial zeros taken in conjugate pairs and in order of imaginary part), and where $$R(s) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \operatorname{li}(s^{1/n}).$$
See, for example, http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/encoding1.htm. 
I am wondering if someone can help by providing a proof of this explicit formula for $\pi_0(x)$ (perhaps from some other known explicit formula, e.g., for $\Pi_0(x)$), or else supply a reference that proves it.  I can't find a proof of it in the literature.