Multiplicative and Additive Arithmetic Functions and Formal Power Series

The theory of arithmetic functions and the theory of formal power series are classical and active parts of mathematics. Algebraic operations on sets of arithmetic functions, called convolutions, have an important place in the theory of arithmetic functions. The theory of formal power series also has its place firmly anchored in abstract algebra. A first goal of this thesis will be to present a parallelism of known characterizations of the concepts of multiplicative and additive for arithmetic functions (Theorems 2.1.2 and 2.2.3) on the one hand and for formal power series on the other (Theorems 3.4.3 and 3.4.4). Therefore, in Chapter 1 and in the first part of Chapter 3 are listed notions and properties that make possible the transposition from arithmetic functions to formal power series. Further, an approach in which formal power series brought to the fore (Sections 3.1 and 3.2) will add new elements in our study on multiplicative arithmetic functions (Section 3.3). So, if mainly, our presentation of Section 3.3 follows P.J. McCarthy’s book [6], the proofs of some main results on completely and specially multiplicative functions has been replaced with new proofs (Theorems 3.2.6, 3.2.7, 3.3.3, 3.3.4) using Bell series. This was a second goal of giving new proofs using Bell series, and so we bring the two topics (arithmetic functions and formal power series) closer together. If in the achievement of the first goal a significant role was played by the embedding of the ring of formal power series in the unitary ring of arithmetic functions, in the case of the second goal, Theorem 2.24 and 2.25 of T.M. Apostol’s book [1] influenced me to use the Bell series in proofs