TITLE: Asymptotic expression for sum of first n prime numbers?
QUESTION [10 upvotes]: Is one known? If not, what are the best known bounds? Is there reason to think that an asymptotic expression is beyond current methods if none exists?

REPLY [7 votes]: Maybe it is interesting to see a easy proof of the simplest approximation. From the PNT and the Abel's summation we have $$\sum_{k=1}^{n}p_{k}\sim\sum_{k=1}^{n}k\log\left(k\right)$$
$$\sum_{k=1}^{n}k\cdot\log\left(k\right)
=\frac{n\left(n+1\right)}{2}\log\left(n\right)-\frac{1}{2}\int_{1}^{n}\frac{\left\lfloor t\right\rfloor \left(\left\lfloor t\right\rfloor +1\right)}{t}dt
 $$ $$=\frac{n^{2}}{2}\log\left(n\right)+O(n^{2})
 $$ where $\left\lfloor t\right\rfloor$ is the floor function, hence

$$\color{red}{\sum_{k=1}^{n}p_{k}\sim\frac{n^{2}}{2}\log\left(n\right).}$$