Arithmetic functions: an algebraic approach.

Abstract

The purpose of this thesis is to investigate arithmetic functions from an algebraic point of view. The emphasis is on algebraic structure on the set of arithmetic functions under two different convolution-type product operations, to Dirichlet products (X), and the unitary product (o). This algebraic approach has the advantage that leads to the development of many classical results in number theory without difficulties and unpleasant computations techniques. The set of arithmetic functions with respect to ordinary addition and Dirichlet product forms unique factorization. However, contrary to Dirichlet product, the set of arithmetic functions with ordinary addition and unitary product is not even in integral domain. Important arithmetic functions and their unitary will be discussed.