Problems on Chapter 1 15
is nonnegative for all real t, so that its discriminant is ≤ 0, that is,
4
k
i=1
aibi
2
− 4
k
i=1
ai
2
k
i=1
bi
2
≤ 0,
which is precisely what needed to be proved.
Problems on Chapter 1
Problem 1.1. Use Proposition 1.1 to prove that if α ≥ 0, then
n≤x
nα
=
xα+1
α + 1
+ O
(xα)
.
Problem 1.2. Prove that
lim
N→∞
13
+
23
+
33
+
43
+ · · · + N
3
N 4
=
1
4
.
Problem 1.3. Use relation ( 1.1) to prove that if α 0, then
n≤x
nα
log n =
xα+1
α + 1
log x −
1
α + 1
+ O
(xα
log x) .
Problem 1.4. Show that the following two representations of the Euler
constant γ are actually the same:
lim
N→∞
N
n=1
1
n
− log N and 1 −
∞
1
t − t
t2
dt.
Problem 1.5. Let f : N → C be a function for which there exists a positive
constant A such that lim
x→∞
1
x
n≤x
f(n) = A. Prove that
n≤x
f(n) log n = A(1 + o(1))x log x (x → ∞).
Problem 1.6. Let f : [a, b] → R be a function which is continuous at x = a.
Define g : [a, b] → R by
g(x) =
0 if x = a,
1 if a x ≤ b.
Prove that
b
a
f dg = f(a).
