a(n) = Sum_{x=2..n} x/log(x) closely approximates the number of primes < n^2.
In fact, the sum is as good as Li(n^2) but summing a(n) is rather time consuming for large n.
For n = 10^9,
a(n) = 24739954333817884,
Pi(n^2) = 24739954287740860,
Li(n^2) = 24739954309690415,
R(n^2) = 24739954284239494,
where Li = Logarithmic integral approximation of Pi, and R = Riemann's approximation of Pi.
Now x/(log(x)1) is a much better approximation of Pi(x) than x/log(x):
10^18/(log(10^18)1) = 24723998785919976,
10^18/log(10^18) = 24127471216847323.
Ironically, though, a(n) = Sum_{x=2..n} x/(log(x)1) is far from Pi(n^2).
