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.)