Regular Convolution On Algebraic Integer Rings

Let N*= ｛1,2,3,… } be the set of positive integers,and RN*the set of all arithmetical functions f:N*→-R,where R is a commutative ring with an identity element.In 1963,W.Narkiewicz introduced a convolution A on the set RN*with the definition and equivalent conditions of a regular convolution.In 1978,V.S.Ramaiah got several properties associated with regular convolutions on the ring of rational integers.On the basis of previous scholars,we study the regular convolutions on the ring of algebraic integers.This paper is organized as follows:In Chapter 1,we introduce the background and development of regular convolutions,at the same time,we give some main results of this paper.In Chapter 2,we review several definitions and properties about regular convolutions on the ring of rational integers.Furthermore,we introduce a generalized Euler totient function.In Chapter 3,we extend the classical regular convolutions on the ring of rational integers to the ring of algebraic integers OKk We introduce a convolution A and a generalized Euler totient function on IoK,which IoK is the set of all non-zero ideals of oK.Then we generalize the well-known Menon’s identity to oK.