Unique factorization of pseudo-arithmetic functions

Abstract

In 1959, Cashwell and Everett proved that the set of complex-valued arithmetic functions forms a unique factorization domain under addition and convolution. In this thesis we further this result in two directions. In the first direction, we replace the set of all natural numbers and the complex field by an arithmetical semigroup S and a unique factorization domain D, respectively, and prove that when the ring of formal power series in any finite number of indeterminates over D is a unique factorization domain, then the set of all pseudo-arithmetic functions from S into D is a unique factorization domain under addition and convolution. In the latter direction, we replace the complex field by a unique factorization ring R with zero divisors and prove that when the ring of formal power series over R in any finite number of indeterminates is a unique factorization ring with zero divisors and the ring of formal power series over R in a countably infinite number or indeterminates is compact, then the set of all arithmetic functions over R is a unique factorization ring with zero divisors under addition and convolution. The proofs employed come from a detailed analysis of those used by cashwell and everett (1959), and lu (1965) with a number of modifications together with an introduction of concepts such as weak cancellation law and compactness