1.4. Exercises 15
by
L(s, χn) =

a=1
χn(a)
as
.
We are ready to write down the formula for S3(n), due to Gauss,
Dirichlet and Shimura (there are also formulas for S5(n), due to Eisen-
stein, Smith, Minkowski and Shimura, and a formula for S7(n), also
due to Shimura). For simplicity, let us assume that n is odd and
square free (for the utmost generality, please check [Shi02]):
S3(n) =
0 if n 7 mod 8,
24

n
π
L(1,χn) otherwise.
The reader is encouraged to investigate this problem further by at-
tempting Exercises 1.4.6 and 1.4.7.
1.4. Exercises
Exercise 1.4.1. Use the divisibility properties of integers to show
that the only solutions to
y2
= x(x + 1)(x + 2) with x, y Z are
(0,0), (−1,0) and (−2,0). (Hint: If a and b are relatively prime and
ab is a square, then a is a square and b is a square.)
Exercise 1.4.2. Find all the Pythagorean triples (a, b, c), i.e., a, b, c
Z and
a2
+
b2
=
c2,
such that
b2
+
c2
=
d2
for some d Z. In other
words, find all the integers a, b, c, d such that (a, b, c) and (b, c, d)
are both Pythagorean triples. (Hint: You may assume that y2 =
x(x + 1)(x + 2) has no rational points other than (0,0), (−1,0) and
(−2,0).)
Exercise 1.4.3. Prove Proposition 1.1.3; i.e., show that f((a, b, c)) is
a point in En, that g((x, y)) is a triangle in Cn and that f(g((x, y))) =
(x, y) and g(f((a, b, c))) = (a, b, c).
Exercise 1.4.4. Calculate S4(n), for n = 1,3,5,6, by hand, using
Jacobi’s formula and also by finding all possible ways of writing n as
a sum of 4 squares.
Exercise 1.4.5. The goal of this problem is to find the q-expansion
of
Θ6(q):
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