16 1. Introduction
(1) Find by hand the values of S6(n), for n = 0,1,2; i.e., find
all possible ways to write n = 0,1,2 as a sum of 6 squares.
(2) Using Sage, calculate the dimension of M
k
2
(Γ1(4)) (see Ap-
pendix A.2) and a basis of modular forms for k = 6.
(3) Write
Θ6
as a linear combination of the basis elements found
in part 2.
(4) Use part 3 to write the q-expansion of Θ6 up to O(q20).
(5) Use the expansion of Θ6 to verify that S6(4) = 252. Also,
calculate S6(19) using Jacobi’s formula and verify that it
coincides with the coefficient of Θ6 in front of the q19 term.
Exercise 1.4.6. Show that any integer n 7 mod 8 cannot be rep-
resented as a sum of three integer squares.
Exercise 1.4.7. Find the number of representations of n = 3 as a
sum of 3 squares. Then compare your result with the value of the
formula given in Example 1.3.3; i.e., use a computer to approximate
S3(3) =
24

3
π
L(1,χ3) =
24

3
π

a=1
(
−3
a
)
a
by adding the first 10,000 terms of L(1,χ3). Do the same for n = 5
and n = 11. Does the formula seem to work for n = 2? (Note:
command kronecker(-n,m) calculates the Kronecker symbol
(
−n
m
)the
in
Sage.)
Exercise 1.4.8. Prove that the Riemann zeta function ζ(s) =
∑∞
n=1
1
ns
has an Euler product; i.e., prove the following formal equality of series

n=1
1
ns
=
p prime
1
1 p−s
.
(Hint: There are two possible approaches:
Hint (a). Expand the right-hand side using the Fundamental Theorem
of Arithmetic and the algebraic equality
1
1+x
=
∑∞
k=0
xk.
[This approach helps build an intuition about what is going
on, but may be hard to write into a rigorous proof]
Hint (b). Calculate (1
1/2s)ζ(s)
and (1
1/3s)(1

1/2s)ζ(s),
etc.)
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