Let $\chi$ be a non-principal Dirichlet character of modulus $q$. The classical Pólya–Vinogradov theorem states that $$\sum_{n \leq x} \chi(n) \leq q^{1/2} \log q.$$ Let now $K$ be a number field of degree $d$ and $\chi$ a finite order non-principal Hecke character of modulus $\mathfrak{q}$. A version of Pólya–Vinogradov here is known, and goes back to Landau 1. It states that $$\sum_{N(\mathfrak{n}) \leq x} \chi(n) \leq (|\Delta_K| N(\mathfrak{q}))^{1/(d+1)}\log(|\Delta_K| N(\mathfrak{q}))^{d} x^{\frac{d-1}{d+1}}.$$ where the sum is over ideals $\mathfrak{n}$ of $\mathcal{O}_K$ and $\Delta_K$ denotes the discriminant of $K$.
There are a few things which surprised me about this result.
Q1: Firstly it is from the 1920's and I couldn't find any improvement in the literature. So are there stronger results available? (I want sums over ideals; I'm not interested in any papers which use sums over integral elements).
Q2: Secondly, I was surprised to see that the upper bound depends on $x$, whereas no such dependence is required over $\mathbb{Q}$. So is this dependence on $x$ really required? Is it possible to replace this bound by $(|\Delta_K| N(\mathfrak{q}))^{A}$ for some $A$ depending on $d$?
1 - E. Landau, Verallgemeinerung eines Pólyaschen Satzes auf algebraische Zahlkörper. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1918).
