On Summatory Functions of Additive Functions and Regular Variation

We shall denote the set of all regularly varying functions by R. We shall also denote by L the set of slowly slowly varying (or slowly oscillating) functions, namely those functions in R for which the index ρ = 0. It is easy to show that if h ∈ R, then there exists L ∈ L such that h(x) = xL(x), with ρ being the index of h. Slowly varying functions arise naturally in many branches of number theory, especially in the theory of arithmetic functions. Namely if a(n) is an arithmetic function, then the summatory function A(x) := ∑