From the Wikipedia article on the prime counting function:
Of profound importance, Bernhard Riemann proved that the prime-counting function is exactly $$\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$$ where $$\operatorname{R}(x)=\sum_{n=1}^\infty \frac{\mu (n)}{n}\operatorname{li}(x^{1/n}),$$ $\mu (n)$ is the Möbius function, $\operatorname{li}(x)$ is the logarithmic integral function, $\rho$ indexes every zero of the Riemann zeta function, and $\operatorname{li}(x^{\rho /n})$ is not evaluated with a branch cut but instead considered as $\operatorname{Ei}\left(\frac{\rho}{n}\ln x\right)$.
From the Mathworld article on the Riemann prime counting function:
Riemann's function is related to the prime counting function by $$\pi (x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho}), \tag*{(12)}$$ where the sum is over all complex (nontrivial) zeros $\rho$ of $\zeta (s)$ (Ribenboim 1996), i.e., those in the critical strip so $0\lt \Re (\rho )\lt 1$, interpreted to mean $$\sum_{\rho}\operatorname{R}(x^{\rho})=\lim_{t\to\infty}\sum_{|\Im (\rho )|\lt t}\operatorname{R}(x^{\rho}). \tag*{(13)}$$ However, no proof of the equality of $(12)$ appears to exist in literature (Borwein et. al 2000).
The last sentence is disputable, as H. M. Edwards provided a proof in 2001 (which is due to Mangoldt, as it seems). But did Riemann prove that equality? I think they would have referenced Riemann or Mangoldt...
User Raymond Manzoni provided a derivation of the equality (Two Representations of the Prime Counting Function):
[...] let's start with a sketch using von Mangoldt's derivation to obtain your equation (1) [the one in this post] that will be used for inspiration [...] This will be the meaning of the ∗ symbols in this article : at a first order discontinuity point (i.e. a jump) the result is the mean value of the limit at the left and the right. [...] we get (with questionable convergence) $$\pi ^{*}(x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^{\rho})$$
Why is the convergence questionable? Is there any proof of the convergence?
