6 1. Preliminary Notions
= A(N)f(N) −
N
1
A(t)f (t)dt
=
n≤N
anf(n),
where the last equality holds since we just proved that (1.4) holds when x
is an integer. Now, since clearly
∑
n≤x
anf(n) =
∑
n≤N
anf(n), the proof
of the proposition is complete.
Before establishing the next result, we introduce an important constant.
Definition 1.5. Let
γ = 1 −
∞
1
t − t
t2
dt.
The number γ is called Euler’s constant. Its numerical value is approxi-
mately 0.57721 . . . .
Remark 1.6. One can easily show that the above definition of the Eu-
ler constant is equivalent to the following one (already mentioned in the
Notation section on page xiii):
γ = lim
N→∞
N
n=1
1
n
− log N
(see Problem 1.4).
It is believed that γ is a transcendental number although it is not even
known if it is irrational. See J. Havil’s excellent book [78] for a thorough
study of this amazing constant.
As an application of this formula we have the following.
Theorem 1.7. For all x ≥ 1,
(1.5)
n≤x
1
n
= log x + γ + O
1
x
.
Proof. Setting an = 1 and f(t) = 1/t in the Abel summation formula, we
easily obtain that
A(x) =
n≤x
1 = x ,
and that f (t) =
−1/t2,
yielding
n≤x
1
n
=
x
x
+
x
1
t
t2
dt
=
x − (x − x )
x
+
x
1
t − (t − t )
t2
dt
