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).
