Connected components of the ranges of twisted divisor functions on number fields

Let $r\in\mathbb{C}$, let $K$ be a finite extension of $\mathbb{Q}$, let $I_K$ be the monoid of integral ideals in the ring of integers $\mathcal{O}_K$ of $K$, and let $\chi$ be a Dirichlet character. Then define the twisted ideal divisor function $\sigma_{r, K, \chi} : I_K \rightarrow \mathbb{C}$ by $$\sigma_{r,K,\chi}(I) = \sum_{J \mid I} N(J)^{-r}\chi(N(J)),$$ where $N$ denotes the ideal norm. For real $r>1,$ we study the number of connected components $C_{r, K, \chi}$ of the closure $\overline{\sigma_{r,K,\chi}(I_K)}$, writing $C_{r,K}$ when $\chi$ is the principal character modulo 1. We prove that $C_{r,K,\chi}$ is finite when $\chi$ is real-valued. When $K = \mathbb{Q}$, we show that for fixed $r > 1,$ every sufficiently large positive integer is realized as $C_{r,\mathbb{Q},\chi},$ and if $r$ is sufficiently large, then every positive integer is realized as $\chi$ varies. For finite Galois extensions $K$ over $\mathbb{Q}$, we exhibit new exponential lower bounds for $C_{r,K},$ and we prove that for every fixed integer $s \geq 2$, the values $C_{r,K}$ are unbounded as $K$ ranges over degree-$s$ extensions of $\mathbb{Q}$.