1.4. Stieltjes integrals 7

= 1 + O

1

x

+

x

1

dt

t

−

x

1

t − t

t2

dt

= 1 + O

1

x

+ log t

t=x

t=1

−

∞

1

t − t

t2

dt −

∞

x

t − t

t2

dt

= log x + γ + O

1

x

+

∞

x

t − t

t2

dt

= log x + γ + O

1

x

+ O

∞

x

dt

t2

= log x + γ + O

1

x

,

as x → ∞, thus completing the proof of the theorem.

1.4. Stieltjes integrals

At times, it is convenient to write certain finite sums as Stieltjes integrals.

Recall that the

Stieltjes1

integral I of the function f over the interval

[a, b] with respect to the function g is defined as

I =

b

a

f dg =

b

a

f(t) dg(t) = lim

Δ→0

n

j=1

f(ξj)(g(tj) − g(tj−1)

with Δ = (t0,...,tn), a = t0, b = tn, Δ = max1≤j≤n(tj − tj−1) and

ξj ∈ [tj−1,tj] for all j = 1,...,n.

If g(x) = x, then

I =

b

a

f(t) d t

is simply the Riemann integral of f. If

g(x) =

0 if a ≤ x ≤ c,

1 if c x ≤ b,

then I = f(c).

For instance, assume that we want to estimate the expression

(1.6)

a≤x

a∈A

f(a),

1Thomas Stieltjes (1850–1894, Holland) was very much interested in elliptic curves and in

number theory. In fact, as Narkiewicz recalls in his book [110], Stieltjes was the first to make

an attempt at a proof of the Riemann Hypothesis. Indeed, he asserted that the series

∞

n=1

μ(n)

ns

converges for all s 1/2, a statement which would have implied the Riemann Hypothesis (see

Problem 3.13 in Chapter 3).