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

TL;DR

This paper extends a Bombieri-type theorem to imaginary quadratic number fields, demonstrating that the Dirichlet convolution of functions with a certain distribution level retains that level in such fields.

Contribution

It proves that the level of distribution is preserved under Dirichlet convolution for arithmetic functions over imaginary quadratic fields with class number one.

Findings

Level of distribution $ heta$ is maintained under convolution.

Applicable to functions on the ring of integers in imaginary quadratic fields.

Extends classical results to a broader algebraic setting.

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 $f, g:\mathcal{O}_K\rightarrow \mathbb{C}$ both have level of distribution $\vartheta$ for some $0<\vartheta\leq 1/2$ then the Dirichlet convolution $f*g$ also have level of distribution $\vartheta$.