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