Convolution On Algebraic Integer Rings

In 1963,Narkiewicz introduced the regular convolution in the integer ring;In 1978,Ramaiah studied a series of related functions and promoted them under the definition of regular convolution;In 2006,Alkan studied some properties of the multivariate arithmetic function Unitary convolution;In 2017,Haukkanen extended the U-nitary convolution result of multivariate functions to Dirichlet con-volution.Based on the above,this paper generalizes the properties of the convolution of arithmetic functions to algebraic integer rings,and obtains some results.The general framework of this paper is as follows:In Chapter 1,we introduce the background of the rugular convo-lution,Unitary convolution and Dirichlet convolution of multivariate arithmetic functions,at the same time,we give some main results of this thesis;In Chapter 2,we introduce some preliminary knowledge needed for the article,including some basic theorems and main definitions;In Chapter 3,we study some properties and main theorems of regular convolution on algebraic integer rings;In Chapter 4,we study the properties of multivariate Dirichlet convolution and Unitary convolution on algebraic integer rings,and obtain some important theorems.