Bombieri-type theorem for convolution of arithmetic functions on Number field

Abstract

Let $K$ be an imaginary quadratic number field of class number one and ${\mathcal{O}}_K$ be its ring of integers. We show that, if the arithmetic functions $f,g:{\mathcal{O}}_K\to \mathbb{C}$ both have level of distribution $\vartheta$ for some $0<\vartheta \le 1/2$ then the Dirichlet convolution $f\ast g$ also have level of distribution $\vartheta$.

10 pages, Comments are welcome