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?

Question

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!

Answer 1

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) 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 $$\frac{1}{y-x}\sum_{x<p\leq y}\log p\to 1.$$