Estimate for sum of negative powers of primes

Question

Specifically, for $a \in (0,1)$, I am interested in the sum $$\sum_{p\leq n} \frac{1}{p^a} $$ as $n$ grows.

Answer 1

See this answer: How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $? Specifically, using partial summation I prove that:

Asymptotic: For $k>-1$ we have $$\sum_{p\leq x}p^{k}=\text{li}\left(x^{k+1}\right)+O\left(x^{k+1}e^{-c\sqrt{\log x}}\right).$$

Where $\text{li}(x)=\int_2^x \frac{1}{\log t}dt$ is the logarithmic integral.

There was also this follow up question:

• Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?

This gets asked a lot on Math.Stackexchange. Here are some of the duplicates:

• Finding an asymptotic for the sum $\sum_{p\leq x}p^m$
• Generalization of the Prime number theorem to $\sum_{p\leq x}p^{m+1}$