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What are the best bounds currently known for the following exponential sum:

$$\sum_{x < p \le 2x} e(\alpha p^k)$$

for values of $\alpha$ far from a rational with small denominator. ($p$ refers to a prime in this case)

In particular, I am interested in such bounds for $k = 9$.

The following gives some bounds: https://projecteuclid.org/download/pdf_1/euclid.mmj/1156345592, but have these bounds been improved since then?

EDIT: I am also interested in such bounds assuming GRH.

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    $\begingroup$ My first idea was to tell you about Kumchev's paper, but then I saw your link exactly leads there. I think you can get the best answer to your question by asking Kumchev himself. $\endgroup$
    – GH from MO
    Jun 18, 2015 at 3:36
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    $\begingroup$ Kumchev has a second paper, arxiv.org/pdf/1112.0201v1.pdf, where he has obtained slight improvements when $k\geq 8$ (see Theorem 4 of that paper). $\endgroup$ Jun 18, 2015 at 8:02

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