Monte-Carlo Approximation of the Prime Counting Function
This is a numerical study of the dependence of the error $\delta\pi(x)$ in the prime counting function when $N$ zeros $\rho$ of the zeta function are used in the sum $\sum_\rho {\rm Li}\,(x^\rho)$.
Figure 3 shows that for $x=10^{12}$ the error $\delta\pi(x)$ is about 1100 for $N=10^3$ and 300 for $N=10^5$. The error drops by roughly a factor of two when the number of zeros is increased by a factor of ten. More extensive numerical data indicates that to reach a sub-unit accuracy in $\pi(x)$ one needs to include about $x^{1.38}$ zeros.