Orbit quotients in the radical ring of multiplicative arithmetic functions
Emil Daniel Schwab
Received February 17, 2025. Published online October 13, 2025.
Abstract: We extend the role of algebraic structures to the study of multiplicative arithmetic functions by associating the corresponding radical ring and a left-group action with the quasi-field (two-sided brace) of multiplicative arithmetic functions. The orbit quotient of two compatible multiplicative arithmetic functions, defined using the orbit set of these functions, brings a new approach to expressing the fundamental properties of multiplicative arithmetic functions. In the last section, we show that some of the basic results concerning Fibonacci and Lucas numbers can be carried over to the quotient setting. Fibonacci numbers connect better with Dirichlet convolution, and Lucas's with unitary convolution. This fact can be perceived within the language united with that of arithmetic functions. The connections between the two integer sequences are highlighted through the two multiplicative prime-independent arithmetic functions, namely the Fibonacci function and the Lucas function. Orbit quotients of these functions and their right-shifted functions are all expressed only with Liouville's lambda function.
Keywords: multiplicative arithmetic function; two-sided brace; radical ring; group action; Fibonacci numbers; Lucas numbers
