I have seen the prime number theorem and on the of versions I know is that $\sum_{p\leq x} \log p=O(x)$. Are there any similar results for say, $\displaystyle \sum_{p\in [x/2, \hspace{0.1em} x]} \log p$ or or $\displaystyle \sum_{p\in [x,y]} \log p$ in general? Also, are there some estimates on $\displaystyle \sum_{p\in[x,y]}\frac{\log p}{p}$ or some $[x,y]$? Perhaps, these are not very hard questions but I don't know how to get good bounds without counting up to $y$ and subtracting from it the count up to $x$. Thank you!