Riemann's Formula

$$J(x)=\mathop{\rm Li}\nolimits (x)-\sum \mathop{\rm Li}\nolimits (x^\rho)+\ln 2\int_x^\infty {dt\over t(t^2-1)\ln t},$$

where $\mathop{\rm Li}\nolimits (x)$ is the Logarithmic Integral, the sum is taken over all nontrivial zeros $\rho$ (i.e., those other than $-2$, $-4$, ...) of the Riemann Zeta Function $\zeta(s)$, and $J(x)$ is Riemann Weighted Prime-Power Counting Function.

See also Logarithmic Integral, Prime Number Theorem, Riemann Weighted Prime-Power Counting Function, Riemann Zeta Function