Let $$K$$ be an imaginary quadratic number field and $$\mathcal{O}_K$$ be its ring of integers. We show that, if the arithmetic functions $$f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}$$ both have level of distribution $$\vartheta$$ for some $$0<\vartheta\leq 1/2$$ then the Dirichlet convolution $$f*g$$ also has level of distribution $$\vartheta$$ . As an application we also obtain an analogue of the Titchmarsh divisor problem for product of two primes in imaginary quadratic fields.

中文翻译：

---

卷积的 Bombieri 型定理在数域上的应用

令 $$K$$ 是一个虚二次数域，$$\mathcal{O}_K$$ 是它的整数环。我们证明，如果算术函数 $$f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}$$ 都具有 $$\vartheta$$ 的分布水平 $$0<\vartheta\leq 1/2$$ 那么狄利克雷卷积 $$f*g$$ 也有分布水平 $$\vartheta$$ 。作为应用，我们还获得了虚二次域中两个素数乘积的 Titchmarsh 除数问题的模拟。