36 1. Elementary Prime Number Theory, I

(d) If e and r are nonnegative integers and r is even, show that there

is an expansion of the form

dr

dxr

(P

(x)e)

=

r

j=r/2

cjj!

e

j

P

(x)e−j

for certain integers cj.

(e) Use the result of part (d) to show that if e is a nonnegative integer

and r ≥ 2n is even, then

1

n!

dr

dxr

(P

(x)e)

is a polynomial in P (x) with

integer coeﬃcients. Conclude that f

(r)(0)

and f

(r)(π)

are integers.

(f) Referring back to definition (1.16), deduce that F (π) + F (0) ∈ Z.

7. In this exercise we present a proof similar to that of J. Hacks (on p. 8)

but relying on the irrationality of π in place of

π2.

Let

χ(n) =

(−1)(n−1)/2

if 2 n,

0 otherwise.

a) Show that χ(n) is a completely multiplicative function, i.e.,

χ(ab) = χ(a)χ(b)

for every pair of positive integers a, b.

b) Assume that there are only finitely many primes. Show that for

every s 0,

∞

n=1

χ(n)

ns

=

p

1 −

χ(p)

ps

−1

.

c) Take s = 1 and obtain a contradiction to the irrationality of π. You

may assume that

π

4

= 1 −

1

3

+

1

5

−

1

7

+ · · · .

8. Say that a natural number n is squarefull if

p2

| n whenever p | n, i.e.,

if every prime showing up in the factorization of n occurs with multi-

plicity 1. Every perfect power is squarefull, but there are many

examples, such as 864 =

25

·

33.

Using Theorem 1.2, show that

∑other

n−1

converges to

ζ(2)ζ(3)

ζ(6)

, where the indicates that the sum is restricted to

squarefull n. Determine the set of real α for which

∑

n−α

converges.

9. (Continuation) Show that every squarefull number has a unique repre-

sentation in the form

u2v3,

where u and v are positive integers with v

squarefree. Deduce that for x ≥ 1,

n≤x

n squarefull

1 =

ζ(3/2)

ζ(3)

x1/2

+

O(x1/3).