Approximation for the sum of primes I have attempted to put together an approximation for the sum of primes.
I've used the much simplified $$\operatorname{li}(x)=\frac{x}{\log(x)-1}$$ combined with $$\frac{x}{2}$$ to give:
$$\frac{x^2}{2(\log(x)-1)}$$ 
The only thing is it is not accurate so:
1) I wonder if I've gone wrong? or does it get accurate with numbers $> 10000$?
2) Suggestions for better approximations but not depending on many iterations.
3) $$x/ \log x$$ pointed out by @mixedmath is the usual way to introduce the prime number theorem but wouldn't using the approximation to $\operatorname{li}(x)$ above be better with $x/2$?

I'm not sure why you're using $x/(\log x - 1)$ instead of $x/ \log x$ (which is what's given in the standard prime number theory), but if you were to use $x/\log x$ you would actually get the correct asymptotic: $x^2 / 2 \log x$. This is a pretty slowly converging asymptotic: the error term is on the order of $O(x^2/\log^2 x)$.
