Groups of Arithmetical Functions

Abstract

Here the sum is taken over all positive integer divisors d of n. This somewhat surprising choice of a product is quite fruitful, allowing one to obtain interesting numbertheoretic formulas from simple computations in the ring. In particular, the useful functions mentioned above can all be expressed in terms of two simple elements of this ring. In this MAGAZINE, Berberian [2] discussed (among other things) the group of units of this ring. He showed that τ , σ , and φ can be expressed in terms of two very simple functions and proved that those two functions are linearly independent. In this article we extend his pair to an uncountably infinite set. In the process, we present answers to other questions posed in his article, including a description of the structure of the group of units. In the interest of accessibility, most of the discussion is confined to real-valued arithmetical functions. Except for a bit of abelian group theory, the algebraic ideas come from introductory linear algebra and abstract algebra. For many readers the only novel concept will be Bell series, a powerful tool developed by E. T. Bell in the early twentieth century.