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Hi there,

Suppose $K$ is a number field. Do we know any bound to the sum(s)?

$\sum_{\substack{\mathbb{N_{n}} \leq x \\ \text{n is an integral ideal of } \mathcal{O_{k}}} }e({ \alpha \mathbb{N_{n}} }) $

where $\alpha$ is any real number.

Or in general

$\sum_{\substack{\mathbb{N_{n}} \leq x \\ \text{n is an integral ideal of } \mathcal{O_{k}}} }e({ h \mathbb{N_{n}^{\theta}} }) $ where $0 <\theta \leq 1$, $h$ is any real number.

($e(x)$=$e^{2 \pi i x}$)

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    $\begingroup$ Does anyone understand this question? $\endgroup$
    – Stefan Kohl ♦
    Commented Jun 5, 2013 at 9:51
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    $\begingroup$ Yes there is a bound given by using $|e(x)| \leq 1$. I'm guessing that you want more than just "any bound", perhaps you could make clear what you want? Do you want to know if you can always do better than this trivial bound if $\alpha \not \in \mathbb{Z}$? If $K=\mathbb{Q}$, the answer to the first question is yes, one has the bound $o(N)$. This is given by the Weyl equidistribution theorem. I would not be surprised if the corresponding result was also known in the number field case, perhaps there is some trick to reduce to the case $K=\mathbb{Q}$. $\endgroup$
    – Daniel Loughran
    Commented Jun 5, 2013 at 13:38

1 Answer 1

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For the first sum, there is a treatment (giving results comparable to those known with $\mathbf{Q}$) in the paper of D. Kane "An asymptotic for the number of solutions to linear equations in prime numbers from specificed residue classes", IJNT 9, no 4, 2013, 1073-1111 (see his Lemma 21).

For the second question, I don't know a reference, but the lemmas and ideas in Kane's paper would certainly help adapting the case of the rationals.

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