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

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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:

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

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