Hello,

I am wondering if there is a simple asymptotic formula for

$$\sum_{p\leq x}\frac{\left(\log p\right)^{k}}{p},$$

where $k\geq0$ is some integer. If $k$ is $0,$ by using the Prime Number Theorem we have

$$\sum_{p\leq x}\frac{1}{p}=\log \log x+b+O\left(e^{-c\sqrt{\log x}}\right).$$

Similarly, the prime number theorem and integration by parts solves the case $k=1$ and gives

$$\sum_{p\leq x}\frac{\left(\log p\right)}{p}=\log x+C+O\left(e^{-c\sqrt{\log x}}\right).$$

My question is do these integrals have a nice asymptotic formula for every $k$? Specifically, I mean with an error term of the form $O\left(e^{-c\sqrt{\log x}}\right).$

Thanks!

Remark: This question is related, and in particular if it is solved with a nice enough asymptotic, then so is this. (but not vice versa)