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.

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