Arithmetical Functions : Infinite Products

TL;DR

This paper introduces classifications of arithmetical functions, explores their properties through polynomials and power series, and establishes a theorem linking these series to infinite products, with applications in probabilistic number theory and Waring's problem.

Contribution

It proposes new classifications of arithmetical functions and establishes a novel theorem connecting arithmetical power series with infinite products.

Findings

Established a theorem relating arithmetical power series and infinite products

Connected arithmetical polynomials to probabilistic number theory

Discussed results related to the Waring problem

Abstract

In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical polynomials and arithmetical power series are introduced. Using these concepts, an interesting Theorem relating arithmetical power series and infinite products has been proved. Also arithmetical polynomials are related to probabilistic number theory. Furthermore some results related to the Waring problem are discussed.