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The best analytic built-in approximation is the Riemann Prime Counting Function; it is implemented in Mathematica as RiemannR. So far we know exact values of $\pi$ prime counting function for n < 10^25, however in Mathematica its counterpart PrimePi[n] can be computed exactly to much lower values i.e. up to 25 10^13 -1, see e.g. What is so special about ...


RiemannR seems to be a better choice than LogIntegral based on this plot: Plot[{PrimePi[n], LogIntegral[n], RiemannR[n]}, {n, 1, 5000}, PlotStyle -> {Black, Blue, Red}] RiemannR[1.*10^1000] 4.344832576401197453*10^996

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