I'm not sure if there is a newer equation, there probably is one from riemman, but Gauss had a formula for approximating the number of primes up to any number x:
[tex]Li(x)=\int_0^x\frac{dt}{log(t)}[/tex]
You could compute this for 10^{100} and then for 10^{101} and subtract the first result from the second and it will give a good estimate.
