We consider a numeration system in the ring of integers $${\mathcal{O}}_K$$ O K of a number field, which we assume to be principal. We prove that the property of being a prime in $${\mathcal{O}}_K$$ O K is decorrelated from two fundamental examples of automatic sequences relative to the chosen numeration system: the Thue–Morse and the Rudin–Shapiro sequences. This is an analogue, in $${\mathcal{O}}_K$$ O K , of results of Mauduit-Rivat which were concerned with the case $$K= \mathbb{Q}$$ K = Q .

中文翻译：

---

主数域中素数处的 Thue-Morse 和 Rudin-Shapiro 序列

我们考虑一个数字域的整数环 $${\mathcal{O}}_K$$ OK 中的计数系统，我们假设它是主体。我们证明了 $${\mathcal{O}}_K$$ OK 中素数的性质是从与所选计算系统相关的自动序列的两个基本示例去相关的：Thue-Morse 和 Rudin-Shapiro 序列。在 $${\mathcal{O}}_K$$ OK 中，这是 Mauduit-Rivat 与 $$K= \mathbb{Q}$$ K = Q 情况有关的结果的模拟。