Generalization of Möbius-Type Arithmetic Functions and Their Algebraic Properties

Abstract

This study aims to formulate a parameterized family of Möbius-type arithmetic functions and investigate their algebraic properties, including multiplicativity, Dirichlet inverses, convolution identities, generalized Möbius inversion formulas, Dirichlet-series representations, and relationships with other arithmetic functions. The research employs a qualitative approach based on a literature review and mathematical-theoretical analysis using formal definitions, deductive proofs, prime-factorization techniques, and Dirichlet convolution theory. The results demonstrate that the proposed function, , preserves multiplicativity, satisfies the reduction principle to the classical Möbius function when , becomes the characteristic function of square-free integers when , and serves as the Dirichlet inverse of . Furthermore, the study derives a generalized Möbius inversion formula, establishes a new family of generalized totient functions, and obtains consistent Dirichlet-series representations. These findings provide an integrated mathematical framework for understanding generalized Möbius-type functions and extend the theoretical foundations of arithmetic identities, inversion theory, and future research in analytic number theory and the algebra of arithmetic functions.