Arithmetical summability

The notation ∑k∣nf(k)∑k∣nf(k) means the finite sum of all numbers f(k)f(k) as kk ranges over the integers that divide nn including 11 and nn. The purpose of this paper is to define a theory of summability suggested by this type of sum and an associated theory of convergence. This paper studies five sequence spaces that are related to the arithmetical summability thus defined.