1.2. Modular forms 9
and S2(9) = 4. Indeed, the number nine has 4 different representa-
tions: 9 = (±3)2 + 02 = 02 + (±3)2. Let us explore other similar
questions.
Let n 0 and k 2. Is the number n 0 a sum of k (integer)
squares? In other words, are there a1,...,ak Z such that n =
a1
2
+···+ak?
2
And if so, in how many different ways can you represent
n as a sum of k squares? Lagrange showed that every natural number
can be represented as a sum of k 4 squares, but how many different
representations are there?
Let Sk(n) be the number of representations of n as a sum of k
squares. Determining exact formulas for Sk(n) is a classical problem
in number theory. There are exact formulas known in a number of
cases (e.g. Eq. 1.2). The formulas for k = 4,6 and 8 are due to
Jacobi and Siegel. We write n =
2νg,
with ν 0 and odd g 0:
S4(n) = 8
d|n, 4 d
d,
S6(n) =
−1
g
22ν+4
4
d|g
−1
d
d2,
S8(n) = 16 ·

d3 if n is odd,
∑d|n
d|n
d3
2

d|g
d3
if n is even.
For example, S4(4) = 8(1 + 2) = 24 and, indeed
4 =
(±1)2
+
(±1)2
+
(±1)2
+
(±1)2
=
(±2)2
+ 0 + 0 + 0
= 0 +
(±2)2
+ 0 + 0 = 0 + 0 +
(±2)2
+ 0 = 0 + 0 + 0 +
(±2)2.
So there are 16 + 2 + 2 + 2 + 2 = 24 possible representations of the
number 4 as a sum of 4 squares. Notice that S4(2) = S4(4). In how
many ways can 4 be represented as a sum of 6 squares? We write
4 =
22
· 1, so ν = 2 and g = 1, and thus,
S6(4) =
−1
1
22·2+4
4
−1
1
·
12
=
(28
4) · 1 = 252.
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