# Estimate for sum of negative powers of primes [duplicate]

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

## 1 Answer

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:

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