5. (Euler) In courses in complex analysis, it is often proved that sin x pos-
sesses the following Weierstrass factorization (valid for all x ∈ C):
(1.15) sin x = x
see, e.g., [Pri01] for a short, direct proof of this identity. A proof using
only real-variable methods appears in [Kob84, Chapter II].
(a) Starting from (1.15), show that
x cot x = 1 − 2
where ζ denotes the Euler-Riemann zeta function. Hint: Take the
logarithmic derivative of both sides.
(b) Computing by hand the first few coeﬃcients in the Taylor series for
x cot x about x = 0, check that ζ(2) =
and ζ(4) =
6. (J. D. Dixon) We outline Dixon’s proof [Dix62] that π is not the root
of a polynomial over Z of degree ≤ 2. The method is that employed
by Niven to show π is irrational (see [Niv47]). Suppose for the sake of
contradiction that π is a root of P (T) = aT
+ bT + c, where a, b and c
are integers, not all vanishing.
Given a polynomial f(T) ∈ R[T], define
(1.16) F (T) := f(T) − f
+ · · · .
Then F (T) ∈ R[T]. View F as a function of a real variable x.
(a) Check that
F (x) sin x − F (x) cos x
= f(x) sin(x),
and conclude that
f(x) sin x dx = F (π) + F (0).
(b) With n a positive integer to be chosen shortly, let f be the polyno-
(T) − P
Show that the left-hand side of (1.17) is strictly between 0 and 1 if
n is suﬃciently large.
We now fix such an n and derive a contradiction by showing that
the right-hand side of (1.17) is an integer.
(c) Show that f
= 0 for all 0 ≤ r 2n.