As a prospective undergraduate who has really benefited from his time on MSE thus far, i recently learnt that there exists asymptotic approximations for $\sum_{p\leq x} 1, \sum_{p\leq x} p, \sum_{p\leq x} \frac{1}{p}, \sum_{p\leq x} \frac{\log p}{p}, \sum_{p\leq x} \log p, \sum_{p^m\leq x, m\geq 1} \log p$ where $p$ is a prime, but have never seen any for $\sum_{p\leq x} p$ ?
Out of curiosity, i'm wondering if there is any such formula yet ?
An idea that quickly came to mind was, $\sum_{p\leq x} p$ = $\sum_{p\leq x} (p/\log p)\log p$, and then apply the Abel/Euler summation theorem ?
$$\log \zeta(s-1) = \sum_{p^k} \frac{p^k}{k } p^{- sk}$$ its abscissa of convergence is $2$ and by the prime number theorem $\log \zeta(s-1) + \log(s-2)$ has a lower abscissa of convergence so $$\sum_{p^k \le x} \frac{p^k}{k}\sim \sum_{p \le x} p \sim \frac{x^2}{2\log x}$$