Lambert series and the four squares theorem

By evaluating the Fourier series at specific points, [Jacobi1829, §40-46] and by playing a few more specialized games, Jacobi was able to obtain Lambert series, series of rational functions in $q$. For example, evaluating the series for dn at $z=0$ gives the series:

$$\frac{2{\bf K}}{\pi} = 1 + 4 \sum \frac{q^n}{1+q^{2n}}$$

Similar tricks give the following Lambert series:

$$\left(\frac{2{\bf K}}{\pi}\right)^2 = 1 + 8 \sum \frac{q^n}{\left(1+(-q)^n\right)^2}$$

These two series are especially important for our purposes because it happens that

$$\vartheta_3(0,q) = \sqrt{\frac{2{\bf K}}{\pi}}$$

In short, the first is related to the two squares problem and the second to the four squares problem.

Furthermore, both the series are easily expanded into power series.

Expanding the first into a power series, Jacobi found that:

$$\frac{2{\bf K}}{\pi} = 1 + 4 \sum_{n=1}^{\infty} (d_1(n)-d_3(n)) q^n$$

Here $d_1(n)$ counts the number of divisors of $n$ which are congruent to 1 modulo 4. Similarly $d_3(n)$ counts the number of divisors of $n$ congruent to 3 modulo 4.

This gives the following theorem:

Theorem (Jacobi's Two Square Theorem). The number of representations of a positive integer as a sum of two squares is equal to four times the difference of the numbers of divisors congruent to 1 and 3 modulo 4.

For example, the divisors of 5 are 5 and 1. Both are congruent to 1 modulo 4. Thus $d_1(5)=2$ and $d_3(5)=0$. Hence 5 has eight representations as the sum of two squares. (These were listed earlier.)

To take a more complicated example, the odd divisors of 75 are 1, 3, 5, 15, 25 and 75. So $d_1(75)=d_3(75)=3$. Hence 75 cannot be represented as the sum of two squares.

Expanding the second Lambert series into a power series gives a series of the form:

$$\left( \frac{2{\bf K}}{\pi} \right)^2 = 1 + 8 \sum_{n=1}^{\infty} \sigma(n:4 \not\vert d) q^n$$

The notation

$$\sigma(n: 4 \not\vert d)$$

is nonstandard notation which means the sum of those divisors of $n$ which are not divisible by 4.

In words, we have Jacobi's Four Square Theorem:

Theorem (Jacobi's Four Square Theorem) The number of representations of an integer as the sum of four squares is equal to eight times the sum of all its divisors which are not divisible by 4.

For example, the divisors of 4 are 1, 2 and 4. So 4 has 8 times (1+2) or 24 representations. Let's check that out. We know that

$$4 = 1^2 + 1^2 + 1^2 + 1^2$$

We obtain $16=2^4$ representations from this using $1^2 = (-1)^2$. In addition,

$$4 = 2^2 + 0^2 + 0^2 + 0^2$$

There are two choices of sign since $2^2=(-2)^2$, but four different ways of ordering the addition, giving eight more representations. Hence, 4 has $24=16+8$ distinct representations as the sum of four squares, as predicted by Jacobi's formula.

The divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100. Of these, 1, 2, 5, 10, 25 and 50 are not divisible by 4. Hence Jacobi's formula predicts

$$8(1+2+5+10+25+50) = 8(93) = 744$$

or 744 distinct ways of writing 100 as a sum of four squares.

To obtain Lagrange's Theorem, first observe that 1 is a divisor of every positive integer and 1 is not divisible by 4. Hence every positive integer has at least eight representations as a sum of four squares.

Corollary (Lagrange's Four Square Theorem). Every positive integer may be represented as the sum of four squares.