10 1. Introduction

The formulas for Sk(n) given above are derived using the theory of

modular forms, as follows. We define a formal power series Θ(q) by

Θ(q) =

∞

j=−∞

qj2

and, for k ≥ 2, consider the power series expansion of the kth power

of Θ:

(Θ(q))k

=

⎛

⎝

∞

j=−∞

qj2

⎞k

⎠

=

∞

a1=−∞

qa1

2

· · ·

∞

ak=−∞

qak

2

=

n≥0

cnqn.

What is the nth coeﬃcient, cn, of

Θk?

If the readers stare at the

previous equation for a while, they will find that cn is given by

cn = #{(a1,...,ak) ∈

Zk

: a1

2

+ · · · + ak

2

= n}.

Therefore, cn = Sk(n) and

(Θ(q))k

=

∑

n≥0

Sk(n)qn.

In other words,

Θk

is a generating function for Sk(n). But, how do we find closed

formulas for Sk(n)? This is where the theory of modular forms be-

comes particularly useful, for it provides an alternative description of

the coeﬃcients of

Θk.

It turns out that, for even k ≥ 2, the function Θk is a modular

form of weight k

2

(more precisely, it is a modular form for the group

Γ1(4)), and the space of all modular forms of weight

k

2

, denoted by

M

k

2

(Γ1(4)), is finite dimensional (we will carefully define all these

terms later). For instance, let k = 4. Then M2(Γ1(4)), the space of

modular forms of weight

4

2

= 2 for Γ1(4), is a 2-dimensional C-vector

space and a basis is given by modular forms with q-expansions:

f(q) = 1 +

24q2

+

24q4

+

96q6

+

24q8

+

144q10

+

96q12

+ · · ·

g(q) = q +

4q3

+

6q5

+

8q7

+

13q9

+

12q11

+

14q13

+ · · · .