Number Theory: An Introduction to Mathematics

542 XIII Connections with Number Theory

by (36) of Chapter XII. Since the theta functions are all solutions of the partial differ-
ential equation

∂^2 y/∂v^2 =− 4 π^2 q∂y/∂q,

the last relation can be written in the form

4 q∂/∂qlog{θ 10 ( 0 )/θ 01 ( 0 )}=θ 004 ( 0 ).

On the other hand, the product expansions of the theta functions show that

θ 10 ( 0 )/θ 01 ( 0 )= 2 q^1 /^4

∏

n≥ 1

( 1 +q^2 n)^2

/∏

n≥ 1

( 1 −q^2 n−^1 )^2

= 2 q^1 /^4

∏

n≥ 1

( 1 −q^4 n)^2

/∏

n≥ 1

( 1 −q^2 n)^2 ( 1 −q^2 n−^1 )^2

= 2 q^1 /^4

∏

n≥ 1

( 1 −q^4 n)^2 ( 1 −qn)−^2.

Differentiating logarithmically, we obtain

θ 004 ( 0 )= 4 q∂/∂qlog{θ 10 ( 0 )/θ 01 ( 0 )}

= 1 + 8

∑

n≥ 1

nqn/( 1 −qn)− 8

∑

n≥ 1

4 nq^4 n/( 1 −q^4 n)

= 1 + 8

∑

n≥ 1

∑

k≥ 1

(nqkn− 4 nq^4 kn)

= 1 + 8

∑

m≥ 1

{σ(m)−σ′(m)}qm,

whereσ(m)is the sum of all positive divisors ofmandσ′(m)is the sum of all pos-
itive divisors ofmwhich are divisible by 4. Since the coefficients in a power series
expansion are uniquely determined, it follows that

r 4 (m)= 8 {σ(m)−σ′(m)}. 

Proposition 1 may also be restated in the form: the number of representations ofm
as a sum of 4 squares is equal to 8 times the sum of the odd positive divisors ofmifm
is odd, and 24 times this sum ifmis even. For example,

r 4 ( 10 )= 24 ( 1 + 5 )= 144.

Since any positive integer has the odd positive divisor 1, Proposition 1 provides a new
proof of Proposition II.40.
The number of representations of a positive integer as a sum of 2 squares may be
treated in the same way, as Jacobi also showed (or, alternatively, by developing further
the arithmetic of Gaussian integers):