Generalized characters over numerical fields and a counterpart of Chudakov hypothesis

The well-known Chudakov hypothesis for numeric characters, conjectured by Chudakov in 1950, suggests that finite-valued numeric character $h(n)$, which satisfies the following conditions: 1) $h(p) \neq 0$ for almost all prime $p$; 2) $S(x) = \sum\limits_{n \leq x} h(n) = \alpha x + O(1)$, is a Dirichlet character. A numeric character which satisfies these conditions is called a generalized character, principal if $\alpha \neq 0$ and non-principal otherwise. Chudakov hypothesis for principal characters was proven in 1964, but for non-principal ones thus far it remains unproved. In this paper we present a definition of generalized character over numerical fields, suggest an analog of Chudakov hypothesis for these characters and provide its proof for principal generalized characters.