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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