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sums are over primes
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user147650
user147650

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 Are there any similar results for say, $\displaystyle \sum_{p\in [x/2, \hspace{0.1em} x]} \log p$$\displaystyle \sum_{\substack{p\in [x/2, \hspace{0.1em} x]\\p \text{ prime}}} \log p$ or or $\displaystyle \sum_{p\in [x,y]} \log p$$\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!

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!

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!

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user147650
user147650

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?

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!