Let $\pi(x)$ be the prime counting function i.e. number of prime numbers less than or equal to $x$.
I know that there are lower bounds and upper bounds to this function but I was wondering if there is any good lower bound or upper bound for partial sums $\sum^{n}_{i=1}\pi(x_i)$ where the $x_i$ are natural numbers. If you know about any book or paper that discussed this matter please let me know.