I have seen the prime number theorem and on the of versions I know is that $\sum\_{p\leq x} \log p=O(x)$ (I am counting over primes here and in the rest of the post). Are there any similar results for say, $\displaystyle \sum\_{\substack{p\in [x/2, \hspace{0.1em} x]\\p \text{ prime}}} \log p$ or or $\displaystyle \sum\_{\substack{p\in [x,y]\\p \text{ prime}}} \log p$ 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!



Are there some results which count $\sum_{p\in [x/2,x]} \log p$ or $\sum_{p\in [x,y]} \log p$ for for $x$ and $y$ positive and real?
Using the various forms of Mertens' theorems and the prime number theorem, you can easily generate bounds for your sums. That is, if you have an approximation for $\sum\_{n\leq y}a(n)$ and $\sum\_{n\leq x}a(n)$, then subtracting one from the other, you get an approximation for $\sum\_{x<n\leq y}a(n)$. However, better results are available by more advanced techniques like zero density estimates for $\zeta(s)$ or sieve methods. For example, [Huxley (1972)](http://www.digizeitschriften.de/download/PPN356556735_0015/PPN356556735_0015___log17.pdf) proved that if $c\in(7/12,1]$ is fixed and $x\to\infty$, then for any $y\in[x+x^c,2x]$ we have
$$\frac{1}{y-x}\sum\_{x<p\leq y}\log p\to 1.$$


