8 1. Preliminary Notions where A ⊂ N and f ∈ C0[1,x]. Let A(x) be the counting function for the set A, that is, let A(x) = #{a ≤ x : a ∈ A}. Then it is easy to see that x 1−0 f(t) d A(t) = n≤x n+0 n−0 f(t) d A(t) = n≤x n∈A n+0 n−0 f(t) d A(t) = n≤x n∈A f(n), which clearly illustrates that the sum (1.6) can be represented by a Stieltjes integral. 1.5. Slowly oscillating functions Very often in number theory, we encounter asymptotic estimates such as π(x) ∼ x log x , p≤x 1 p ∼ log log x, n≤x f(n) = x + O x e √ log x . In these statements, the functions log x, log log x and e √ log x are all of a particular type: they all belong to the class of slowly oscillating functions. Definition 1.8. A function L : [M, +∞) → R continuous on [M, +∞), where M is a positive real number, is said to be a slowly oscillating function if for each positive real number c 0, (1.7) lim x→∞ L(cx) L(x) = 1. We denote by L the set of slowly oscillating functions. It is possible to show (see Seneta [128]) that a differentiable function L belongs to L if and only if (1.8) xL (x) L(x) = o(1) (x → ∞) and, in fact, that L ∈ L if and only if there exists x0 0 such that L(x) = C(x) exp x x0 η(t) t dt , where limx→∞ C(x) = C, for a certain constant C = 0, and where η(t) is a function which tends to 0 as t → ∞. This last result is often called the Representation theorem for slowly oscillating functions. A function R : [M, +∞) → R continuous on [M, +∞), where M is a positive real number, is said to be regularly varying if there exists a real

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