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).
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