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
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
is a polynomial in P (x) with
integer coeﬃcients. Conclude that f
(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
if 2 n,
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,
c) Take s = 1 and obtain a contradiction to the irrationality of π. You
may assume that
= 1 −
+ · · · .
8. Say that a natural number n is squarefull if
| 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 =
Using Theorem 1.2, show that
, where the indicates that the sum is restricted to
squarefull n. Determine the set of real α for which
9. (Continuation) Show that every squarefull number has a unique repre-
sentation in the form
where u and v are positive integers with v
squarefree. Deduce that for x ≥ 1,