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).