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