18.781, Fall 2007 Problem Set 11
Due: WEDNESDAY, November 28
These problems all relate to the proof that there are infinitely many
primes in an arithmetic progression. While this is covered in Niven, Zucker
man, and Montgomery, our proof differs from theirs and notes are available
in the ”Solutions and Handouts” section of the course webpage.
1. We will say that an infinite product
∞
n
=1
a
n
converges if the limit of the partial products
P
N
=
N
n
=1
a
n
converges as
N
→ ∞
and is not 0.
(That’s the first approximation
to the definition. In practice, we want to allow a finite number of the
factors
a
n
in the product to be 0. Suppose
N
0
is the largest index with
a
N
0
= 0. Then we will say the product converges if
P
N
=
N
n
=
N
0
+1
a
n
converges as
N
→ ∞
.)
(a) Give an example of an infinite product whose partial products
converge to 0. (Hence, we don’t regard it as an infinite product.)
(b) Show that any convergent infinite product can be written in the
form
∞
n
=1
(1 +
b
n
)
with
lim
n
→∞
b
n
= 0
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 Spring '09
 BRUBAKER
 Number Theory, Arithmetic progression, Riemann zeta function, Euler product, infinite product

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